Solving parametric systems of polynomial equations over the reals through Hermite matrices
aa r X i v : . [ c s . S C ] N ov S OLVING PAR AMETRIC SYSTEMS OF POLYNOMIAL EQUATIONSOVER THE R EALS THROUGH H ER MITE MATR IC ES
Huu Phuoc Le
Sorbonne Université, CNRS,Laboratoire d’Informatique de Paris 6, LIP6,Équipe P OL S YS F-75252, Paris Cedex 05, France [email protected]
Mohab Safey El Din
Sorbonne Université, CNRS,Laboratoire d’Informatique de Paris 6, LIP6,Équipe P OL S YS F-75252, Paris Cedex 05, France [email protected]
December 1, 2020 A BSTRACT
We design a new algorithm for solving parametric systems of equations having finitely many com-plex solutions for generic values of the parameters. More precisely, let f = ( f , . . . , f m ) ⊂ Q [ y ][ x ] with y = ( y , . . . , y t ) and x = ( x , . . . , x n ) , V ⊂ C t × C n be the algebraic set defined by the si-multaneous vanishing of the f i ’s and π be the projection ( y , x ) → y . Under the assumptions that f admits finitely many complex solutions when specializing y to generic values and that the idealgenerated by f is radical, we solve the following algorithmic problem. On input f , we compute semi-algebraic formulas defining open semi-algebraic sets S , . . . , S ℓ in the parameters’ space R t such that ∪ ℓi =1 S i is dense in R t and, for ≤ i ≤ ℓ , the number of real points in V ∩ π − ( η ) isinvariant when η ranges over S i .This algorithm exploits special properties of some well chosen monomial bases in the quotient al-gebra Q ( y )[ x ] /I where I ⊂ Q ( y )[ x ] is the ideal generated by f in Q ( y )[ x ] as well as the spe-cialization property of the so-called Hermite matrices which represent Hermite’s quadratic forms.This allows us to obtain “compact” representations of the semi-algebraic sets S i by means of semi-algebraic formulas encoding the signature of a given symmetric matrix.When f satisfies extra genericity assumptions (such as regularity), we use the theory of Gröbnerbases to derive complexity bounds both on the number of arithmetic operations in Q and the degreeof the output polynomials. More precisely, letting d be the maximal degrees of the f i ’s and D = n ( d − d n , we prove that, on a generic input f = ( f , . . . , f n ) , one can compute those semi-algebraic formulas with O e(cid:16)(cid:0) t + D t (cid:1) t n t +1 d nt +2( n + t )+1 (cid:17) arithmetic operations in Q and thatthe polynomials involved in these formulas have degree bounded by D .We report on practical experiments which illustrate the efficiency of this algorithm, both on genericparametric systems and parametric systems coming from applications since it allows us to solvesystems which are out of reach on the current state-of-the-art. K eywords Real algebraic geometry ; Polynomial system solving ; Real root classification; Hermite quadratic forms;Gröbner bases † Mohab Safey El Din and Huu Phuoc Le are supported by the ANR grants ANR-18-CE33-0011 S
ESAME , and ANR-19-CE40-0018 D E R ERUM N ATURA , the joint ANR-FWF ANR-19-CE48-0015 ECARP project, the PGMO grant CAM I SA DO and theEuropean Union’s Horizon 2020 research and innovative training network program under the Marie Skłodowska-Curie grant agree-ment N° 813211 (POEMA). Introduction
In the whole paper, Q , R and C denote respectively the fields of rational, real and complex numbers.Let f = ( f , . . . , f m ) be a polynomial sequence in Q [ y ][ x ] where the indeterminates y = ( y , . . . , y t ) are consideredas parameters and x = ( x , . . . , x n ) are considered as variables . We denote by V ⊂ C t × C n the (complex) algebraicset defined by f = · · · = f m = 0 and by V R its real trace V ∩ R t + n . We consider also the projection on the parameterspace y π : C t × C n → C t , ( y , x ) y . Further, we say that f satisfies Assumption ( A ) when the following holds. Assumption A . There exists a non-empty Zariski open subset
O ⊂ C t such that π − ( η ) ∩ V is non-empty and finitefor any η ∈ O . In other words, assuming ( A ) ensures that, for a generic value η of the parameters, the sequence f ( η, · ) defines a finitealgebraic set and hence finitely many real points. Note that, it is easy to prove that one can choose O in a way thatthe number of complex solutions to the entries of f ( η, · ) is invariant when η ranges over O (e.g. using the theory ofGröbner basis). This is no more the case when considering real solutions whose number may vary when η ranges over O .By Hardt’s triviality theorem (Hardt, 1980), there exists a real algebraic proper subset R of R t such that, for anynon-empty connected open set U of R t \ R and η ∈ U , π − ( η ) × U is homeomorphic with π − ( U ) .This leads us to consider the following real root classification problem. Problem 1 (Real root classification) . On input f satisfying Assumption ( A ) , compute semi-algebraic formulas (i.e.finitely many disjunctions of conjunctions of polynomial inequalities) defining semi-algebraic sets S , . . . , S ℓ suchthat (i) The number of real points in V ∩ π − ( η ) is invariant when η ranges over S i , for ≤ i ≤ ℓ ;(ii) The union of the S i ’s is dense in R t ;as well as at least one sample point η i in each S i and the corresponding number of real points in V ∩ π − ( η i ) .A collection of semi-algebraic formulas sets is said to solve Problem (1) for the input f if it defines a collection ofsemi-algebraic sets S i satisfies the above properties (i) and (ii).Our output will have the form { (Φ i , η i , r i ) | ≤ i ≤ ℓ } where Φ i is a semi-algebraic formula defining the set S i , η i ∈ Q t is a sample point of S i and r i is the corresponding number of real roots. A weak version of Problem (1) would be to compute only a set { η , . . . , η ℓ } of sample points for a collection ofsemi-algebraic sets S i solving Problem (1) and their corresponding numbers of real points in V ∩ π − ( η j ) . Example 2.
Consider the equation x + y x + y = 0 where y and y are the parameters and x is the uniquevariable. While y − y = 0 , this equation always has exactly two distinct complex solutions. On the other hand, itsnumber of distinct real solutions can take any value from to , depending on the sign of the discriminant y − y .One possible output for Problem (1) on this toy example is the following: y − y < , (0 , , real solution y − y = 0 , (2 , , real solution y − y > , (1 , , real solutionsObserve that another possible output is (cid:26) y − y < , (0 , , real solution y − y > , (1 , , real solutionsas the above two inequalities define semi-algebraic sets whose union is dense in R . Problem (1) appears in many areas of engineering sciences such as robotics or medical imagery (see, e.g.,Yang and Zeng (2000); Corvez and Rouillier (2002); Yang and Zeng (2005); Faugère et al. (2008); Bonnard et al.22016)). In those applications, the behavior of mechanisms or complex systems depends on intrinsic parametersthat are related by polynomial equations or inequalities. Thus, the polynomial systems arising from those applicationsare naturally parametric and most of the time the end-user is interested in classifying the number of real roots withrespect to parameters’ values.
A first approach to Problem (1) would be to compute a cylindrical algebraic decomposition (CAD) of R t × R n adaptedto f using e.g. Collins’ algorithm (and its more recent improvements) ; see Collins (1976). While, up to our knowledge,there is no clear reference for this fact, the cylindrical structure of the cells of the CAD will imply that their projectionon the parameters’ space R t define semi-algebraic sets enjoying the properties needed to solve Problem (1). However,the doubly exponential complexity of CAD both in terms of runtime and output size (Davenport and Heintz, 1988;Brown and Davenport, 2007) makes it difficult to use in practice.A more popular approach consists in computing polynomials h , . . . , h r in Q [ y ] such that ∪ ri =1 V ( h i ) ∩ R t contains theboundaries of semi-algebraic sets S , . . . , S ℓ enjoying the properties required to solve Problem (1). Next, one needs tocompute semi-algebraic descriptions of the connected components of R t \ ∪ ri =1 V ( h i ) as well as sample points in theseconnected components. This is basically the approach followed by Yang and Xia (2005) (the h i ’s are called borderpolynomials ) and Lazard and Rouillier (2007) (the set ∪ ri =1 V ( h i ) is called discriminant variety ) under the assumptionthat h f i is a radical ideal. Note that both (Yang and Xia, 2005) and (Lazard and Rouillier, 2007) provide algorithmsthat can handle variants of Problem (1) allowing inequalities. In this paper, we focus on the situation where we onlyhave equations in our input parametric system.That being said, when h f i is radical and the restriction of π to V ∩ R t × R n is proper, one can easily prove usinga semi-algebraic version of Thom’s isotopy lemma (Coste and Shiota, 1992) that one can choose ∪ ri =1 V ( h i ) to bethe set critical values of the restriction of π to V (see e.g. Bonnard et al. (2016)). If f is a regular sequence (hence m = n ), the critical set of the restriction of π to V is defined as the intersection of V with the hypersurface definedby the vanishing of the determinant of the Jacobian matrix of f with respect to the variables x . When d dominatesthe degrees of the entries of f , Bézout’s theorem allows us to state that the degree of this set is bounded above by n ( d − d n .It is worth noticing that, usually, this approach is used only to solve the aforementioned weak version of Problem (1)as getting a semi-algebraic description of the connected components of R t \ ∪ ri =1 V ( h i ) through CAD is too expensivewhen t ≥ (still, because of the doubly exponential complexity of CAD). Under the above assumptions and notation,the output degree of the polynomials in such formulas would be bounded by ( n ( d − d n ) O ( t ) .An alternative would be to use parametric roadmap algorithms to do such computations using e.g. (Basu et al., 2006,Chap. 16) to compute semi-algebraic representations of the connected components of R t \∪ ri =1 V ( h i ) . Under the aboveextra assumptions, this would result in output formulas involving polynomials of degree bounded by ( n ( d − d n ) O ( t ) using ( n ( d − d n ) O ( t ) arithmetic operations (see (Basu et al., 2006, Theorem 16.13)). Note that the output degreesare by several orders of magnitude larger than n ( d − d n which bounds the degree of the set of critical values of therestriction of π to V .Hence, one topical algorithmic issue is to design an efficient algorithm for solving Problem (1) which would outputsemi-algebraic formulas of degree bounded by n ( d − d n . At this stage of our exposition, this is not clear that it isdoable.We describe in detail our contributions in the next paragraph but we can already state that, when f enjoys somegenericity properties that are made clear further, the algorithm we design outputs semi-algebraic formulas involvingpolynomials of degree bounded by n ( d − d n and which are computed using ( n ( d − d n ) O ( t ) arithmetic operationsin Q .To achieve these results, we revisit tools for univariate real root counting, such as Sturm and Sturm-Habicht sequencesand Hermite’s quadratic form to adapt them in our multivariate setting. This leads us to mention González-Vega et al.(1998); Liang et al. (2008) or Henrion (2010) which provide algorithms for classifying the real roots of a univariatepolynomial with coefficients in Q [ y ] , hence restricted to the case where n = 1 (either using Sturm-based techniquesor Hermite’s quadratic forms). 3 .3 Main results We start by revisiting Sturm-based methods in a multivariate context. We basically use the algorithm of Schost (2003)to compute a rational parametrization of V = V ( f ) with respect to the x -variables. More precisely, we computea sequence of polynomials ( w, v , . . . , v n ) in Q ( y )[ u ] where u is a new variable, such that the constructible set Z ⊂ C t × C n of every point (cid:18) η, v ∂w/∂u ( η, ϑ ) , . . . , v n ∂w/∂u ( η, ϑ ) (cid:19) , where ( η, ϑ ) ∈ C t × C such that w ( η, ϑ ) = 0 and η does not cancel ∂w/∂u and any denominator of ( w, v , . . . , v n ) ,is Zariski dense in V , i.e., the Zariski closure of Z coincides with V .Then, using the bi-rational equivalence between Z and its projection on the ( u, y ) -space, we establish that semi-algebraic formulas solving Problem (1) can be obtained through the computation of the subresultant sequence associ-ated to (cid:0) w, ∂w∂u (cid:1) . This is admittedly folklore in symbolic computation but, as far as we know, is not explicitly writtenin the literature. In particular, the analysis of degree bounds derived from this strategy is one of our contributions.Before stating our first complexity result, we need to introduce the complexity model which is used. Throughout thispaper, we measure only the arithmetic complexity of algorithms, i.e., the number of arithmetic operations + , − , × , ÷ ,in the base field Q . We use the Landau notation:• Let f : R ℓ + R + be a positive function. We let O ( f ) denote the class of functions g : R ℓ + → R + such thatthere exist C, K ∈ R + such that for all k x k ≥ K , g ( x ) ≤ Cf ( x ) , where k · k is a norm of R ℓ .• The notation O e denotes the class of functions g : R ℓ + → R + such that g ∈ O ( f log κ ( f )) for some κ > .Further, the notation ω always stands for the exponent constant of the matrix multiplication, i.e., the smallest positivenumber such that the product of two matrices in Q N × N can be done using O ( N ω ) arithmetic operations in Q . Thevalue of ω can be bounded from above by . , which is a recent result established in Le Gall (2014).Under some genericity assumptions on the input system, Theorem I establishes the complexity result of our Sturm-based algorithm and also the degree bound for polynomials involved in the semi-algebraic formulas solving Prob-lem (1) obtained this way. Its proof is given in Section 4, where all the genericity assumptions are clarified. Theorem I.
Let f = ( f , . . . , f n ) ⊂ Q [ y ][ x ] be a generic parametric system and d be the largest total degree amongthe deg( f i ) ’s.Then, there exists a probabilistic algorithm that computes semi-algebraic descriptions of a set of semi-algebraic setssolving Problem 1 within O e(cid:18)(cid:18) t + 2 d n t (cid:19) t d nt +3 n (cid:19) arithmetic operations in Q in case of success.These semi-algebraic formulas computed by this algorithm involve polynomials in Q [ y ] of degree bounded by d n . When reporting experimental results, we will see that, even though the complexity bound we obtain lies in d O ( nt ) , thisapproach does not allow us to solve problems faster than the state-of-the-art. One bottleneck comes from the fact thatthe polynomials of the output semi-algebraic formulas have degree way higher than the bound n ( d − d n which wewill prove to apply under the same assumptions as Theorem I using different algorithmic strategies.Note that the above Sturm-based approach as well as the ones which consist in computing polynomials in Q [ y ] todefine a set discriminating semi-algebraic sets in R t enjoying the properties needed to solve Problem (1) combine twosteps of algebraic elimination. The semi-algebraic formulas are obtained through intermediate data who have beenobtained through an elimination step.The rest of the paper then focuses on an alternative approach which computes semi-algebraic formulas solving Prob-lem (1) by avoiding interlaced algebraic elimination steps. We will see (as announced earlier) that under genericityassumptions, this allows us to obtain a degree bound and an arithmetic cost which are better than the Sturm-basedalgorithm by one order of magnitude.To do that, we rely on well-known properties of Hermite quadratic forms to count the real roots of zero-dimensionalideals ; see (Hermite, 1856). Basically, given a zero-dimensional ideal I ⊂ Q [ x ] , Hermite’s quadratic form operateson the finite dimensional Q -vector space A := Q [ x ] /I as follows A × A → Q h, k ) trace( L h · k ) , where L h · k denotes the endomorphism p h · k · p of A .The number of distinct real (resp. complex) roots of the algebraic set defined by I equals the signature (resp. rank) ofHermite’s quadratic form ; see e.g. (Basu et al., 2006, Theorem 4.102). Recall that such quadratic form is representedby a symmetric matrix of size δ × δ , where δ is the degree of I , once a basis of the finite dimensional vector spaceon which the form operates is fixed. Hence, the signature of a Hermite quadratic form can be computed once a matrixrepresentation, which we call Hermite’s matrix, of this quadratic form is known (Basu et al., 2006, Algo. 8.43).We first slightly extend the definition of Hermite’s quadratic forms and Hermite’s matrices to the context of parametricsystems; we call them parametric Hermite quadratic forms and parametric Hermite matrices. This is easily done sincethe ideal of Q ( y )[ x ] generated by f , considering Q ( y ) as the base field, has dimension zero. We also establish naturalspecialization properties for these parametric Hermite matrices.Hence, a parametric Hermite matrix, similar to its zero-dimensional counterpart, allows one to count respectively thenumber of distinct real and complex roots at any parameters outside a strict algebraic sets of R t by evaluating thesignature and rank of its specialization.Based on this specialization property, we design two algorithms for solving Problem (1) and also its weak version forthe input system f which satisfies Assumption ( A ) and generates a radical ideal.Our algorithm for the weak version of Problem (1) reduces to the following main steps.(a) Compute a parametric Hermite matrix H associated to f ⊂ Q [ y ][ x ] .(b) Compute a set of sample points { η , . . . , η ℓ } in the connected components of the semi-algebraic set of R t defined by w = 0 where w is derived from H .This is done through the so-called critical point method (see e.g. (Basu et al., 2006, Chap. 12) and referencestherein) which are adapted to obtain practically fast algorithms following Safey El Din and Schost (2003).We will explain in detail this step in Section 3.This algorithm takes as input s polynomials of degree D involving t variables and computes sample pointsper connected components in the semi-algebraic set defined by the non-vanishing of these polynomials using O e(cid:18)(cid:18) D + tt (cid:19) (2 t ) s t +1 t D t +1 (cid:19) . (c) Compute the number r i of real points in V ∩ π − ( η i ) for ≤ i ≤ ℓ .This is done by simply evaluating the signature of the specialization of H at each η i .It is worth noting that, in the algorithm above, we obtain through parametric Hermite matrices a polynomial w thatplays the same role as the discriminant varieties of Lazard and Rouillier (2007) or the border polynomials of Yang et al.(2001). We will see in the section reporting experiments that our approach outperforms the other twos on everyexample we consider.To return semi-algebraic formulas, we follow a slightly different routine:(a) Compute a parametric Hermite matrix H associated to f ⊂ Q [ y ][ x ] .(b) Compute a set of sample points { η , . . . , η ℓ } in the connected components of the semi-algebraic set of R t defined by ∧ δi =1 M i = 0 where the M i ’s are the leading principal minors of H . Again, this is done by thealgorithm given in Section 3.(c) For ≤ i ≤ ℓ , evaluate the sign pattern of ( M , . . . , M δ ) at the sample point η i . From this sign pattern, weobtain a semi-algebraic formula representing the connected component corresponding to η i .(d) Compute the number r i of real points in V ∩ π − ( η i ) for ≤ i ≤ ℓ .Another contribution of this paper is to make clear how to perform the step (a). For this, we rely on the theory ofGröbner bases.More precisely, we use specialization properties of Gröbner bases, similar to those already proven in (Kalkbrener,1997). This leaves some freedom when running the algorithm: since we rely on Gröbner bases, one may choosemonomial orderings which are more convenient for practical computations.In particular, the monomial basis of the quotient ring Q ( y )[ x ] /I where I is the ideal generated by f in Q ( y )[ x ] depends on the choice of the monomial ordering used for Gröbner bases computations. We describe the behavior of5ur algorithm when choosing the graded reverse lexicographical ordering whose interest for practical computationsis explained in Bayer and Stillman (1987); Bayer and Stillman (1987). Further, we denote by grevlex ( x ) the gradedreverse lexicographical ordering applied to the sequence of the variables x = ( x , . . . , x n ) (with x ≻ · · · ≻ x n ).Further, we also denote by ≻ lex the lexicographical ordering.We report, at the end of the paper, on the practical behavior of this algorithm. In particular, it allows us to solveinstances of Problem (1) which were not tractable by the state-of-the-art as well as the actual degrees of the polynomialsin the output formula which are bounded by n ( d − d n .We actually prove such a statement under some generic assumptions. Our main complexity result is stated below. Itsproof is given in Subsection 7.2, where the generic assumptions in use are given explicitly. Theorem II.
Let C [ x , y ] d be the set of polynomials in C [ x , y ] having total degree bounded by d and set D = n ( d − d n . There exists a non-empty Zariski open set F ⊂ C [ x , y ] nd such that for f = ( f , . . . , f n ) ∈ F ∩ Q [ x , y ] n ,the following holds:i) There exists an algorithm that computes a solution for the weak-version of Problem (1) within O e(cid:18)(cid:18) t + D t (cid:19) t n t +1 d nt + n +2 t +1 (cid:19) . arithmetic operations in Q .ii) There exists a probabilistic algorithm that returns the formulas of a collection of semi-algebraic sets solvingProblem (1) within O e(cid:18)(cid:18) t + D t (cid:19) t n t +1 d nt +2( n + t )+1 (cid:19) arithmetic operations in Q in case of success.iii) The semi-algebraic descriptions output by the above algorithm involves polynomials in Q [ y ] of degreebounded by D . We note that the binomial coefficient (cid:0) t + D t (cid:1) is bounded from above by D t ≃ n t d nt + t . Therefore, the complexitiesgiven in the items i) and ii) of Theorem II can be bounded by O e(cid:0) t n t d nt (cid:1) and O e(cid:0) t n t d nt (cid:1) respectively. Organization of the paper
This paper is structured as follows. Section 2 reviews fundamental notions of algebraicgeometry and the theory of Gröbner bases that we use further. In Section 4, we discuss an algorithm based onSturm’s theorem for computing semi-algebraic formulas of the set S i . This provides an overview on the drawbacksand potential improvements of this approach. Section 5 lies the definition and some useful properties of parametricHermite matrices. There, we also present an algorithm with some optimizations to compute such a matrix. In Section6, we describe our algorithm for solving the real root classification problem using this parametric Hermite matrix. Thecomplexity analysis of the algorithms mentioned above is given in Section 7. Finally, in Section 8, we report on thepractical behavior of our algorithms and illustrate its practical capabilities. In the first paragraph, we fix some notations on ideals and algebraic sets and recall the definition of critical pointsassociated to a given polynomial map. This notion is the foundation of many algorithms in semi-algebraic geom-etry such as computing sample points of connected components (Bank et al., 2001; Safey El Din and Schost, 2003;Bank et al., 2005), polynomial optimization (Safey El Din, 2008; Guo et al., 2010) or answering connectivity queriesusing roadmaps (Safey El Din and Schost, 2011; Basu and Roy, 2014; Basu et al., 2014; Safey El Din and Schost,2017). Next, we give the definitions of regular sequences, Hilbert series, Noether position and proper maps, which areused later in Subsection 7.1 for the complexity analysis of our algorithms. The fourth paragraph recalls some basicproperties of Gröbner bases and quotient algebras of zero-dimensional ideals. We refer to Cox et al. (2007) for anintroductory study on the algorithmic theory of Gröbner bases. In the last paragraphs, we recall respectively the defini-tions of zero-dimensional parametrizations and rational parametrizations. The zero-dimensional parametrization goesback to Kronecker (1882) and is widely used in computer algebra (see e.g. Gianni and Teo Mora (1987); Giusti et al.(2001, 1995)) to represent finite algebraic sets. In Section 3, the output of our algorithm for computing points perconnected components of the non-vanishing locus of a given set of polynomials is encoded by this parametrization.On the other hand, the rational parametrization, which generalizes the notion of zero-dimensional parametrizations, isintroduced in Schost (2003) as the data representation for the parametric geometric resolution algorithm. In Section 4,6e compute a rational parametrization of the input system using the algorithm of Schost (2003) to reduce Problem (1)to the univariate case.
Algebraic sets and critical points
We consider a sub-field F of C . Let I be a polynomial ideal of F [ x , . . . , x n ] , thealgebraic subset of C n at which the elements of I vanish is denoted by V ( I ) . Conversely, for an algebraic set V ⊂ C n ,we denote by I ( V ) ⊂ C [ x , . . . , x n ] the radical ideal associated to V . Given any subset A of C n , we denote by A theZariski closure of A , i.e., the smallest algebraic set containing A .A map ϕ between two algebraic sets V ⊂ C n and W ⊂ C s is a polynomial map if there exist ϕ , . . . , ϕ t ∈ C [ x , . . . , x n ] such that the ϕ ( η ) = ( ϕ ( η ) , . . . , ϕ s ( η )) for η ∈ V .An algebraic set V is equi-dimensional of dimension t if it is the union of irreducible algebraic sets of dimension t .Let ϕ be a polynomial map from V to another algebraic set W . The morphism ϕ is dominant if and only if the imageof every irreducible component V ′ of V by ϕ is Zariski dense in W , i.e. ϕ ( V ′ ) = W .Let φ ∈ C [ x , . . . , x n ] which defines the polynomial function φ : C n → C , ( x , . . . , x n ) φ ( x , . . . , x n ) and V ⊂ C n be a smooth equi-dimensional algebraic set. We denote by crit( φ, V ) the set of critical points of therestriction of φ to V . If c is the codimension of V and ( f , . . . , f m ) generates the vanishing ideal associated to V , then crit( φ, V ) is the subset of V at which the Jacobian matrix associated to ( f , . . . , f m , φ ) has rank less than or equal to c (see, e.g., (Safey El Din and Schost, 2017, Subsection 3.1)). Regular sequences & Hilbert series
Let F be a field and ( f , . . . , f m ) ⊂ F [ x ] where x = ( x , . . . , x n ) and m ≤ n be a homogeneous polynomial sequence. We say that ( f , . . . , f m ) ⊂ F [ x ] is a regular sequence if for any i ∈ { , . . . , m } , f i is not a zero-divisor in F [ x ] / h f , . . . , f i − i .The notion of regular sequences is the algebraic analogue of complete intersection. In this paper, we focus particularlyon the Hilbert series of homogeneous regular sequences, which are recalled below.Let I ⊂ F [ x ] be a homogeneous ideal. We denote by F [ x ] r the set of every homogeneous polynomial whose degreeis equals to r . Then F [ x ] r and I ∩ F [ x ] r are two F -vector spaces of dimensions dim F ( F [ x ] r ) and dim F ( I ∩ F [ x ] r ) respectively. The Hilbert series of I is defined as HS I ( z ) = ∞ X r =0 (dim F ( F [ x ] r ) − dim F ( I ∩ F [ x ] r )) · z r . When I can be generated by a homogeneous regular sequence ( f , . . . , f m ) , the explicit form of the Hilbert series of I is known (see, e.g., Moreno-Socıas (2003)): HS I ( z ) = Q mi =1 (cid:0) − z deg( f i ) (cid:1) (1 − z ) n . We now consider the affine polynomial sequences. Note that one can define affine regular sequences by simplyremoving the homogeneity assumption of ( f , . . . , f m ) from the above definition. However, as explained in (Bardet,2004, Sec 1.7), many important properties that hold for homogeneous regular sequences are no longer valid for theaffine ones. Therefore, in this paper, we use (Bardet, 2004, Definition 1.7.2) of affine regular sequences, which is morerestrictive but allows us to preserve similar results as the homogeneous case. We recall that definition below.For p ∈ F [ x , . . . , x n ] , we denote by H p the homogeneous component of largest degree of p . A polynomial se-quence ( f , . . . , f m ) ⊂ F [ x , . . . , x n ] , not necessarily homogeneous, is called a regular sequence if and only if ( H f , . . . , H f m ) is a homogeneous regular sequence. Noether position & Properness
Let F be a field and f = ( f , . . . , f n ) ⊂ F [ x , . . . , x n + t ] . The variables ( x , . . . , x n ) are in Noether position with respect to the ideal h f i if their canonical images in the quotient algebra F [ x , . . . , x n + t ] / h f i are algebraic integers over F [ x n +1 , . . . , x n + t ] and, moreover, F [ x n +1 , . . . , x n + t ] ∩ h f i = h i .From a geometric point of view, Noether position is strongly related to the notion of proper map below (see Bardet et al.(2015)).Let V be the algebraic set defined by f ∈ R [ y , . . . , y t , x , . . . , x n ] . The restriction of the projection π : ( y , x ) y to V ∩ R t + n is said to be proper if the inverse image of every compact subset of π ( V ∩ R t + n ) is compact. If the variables7 = ( x , . . . , x n ) is in Noether position with respect to h f i , then the projection π : V ∩ R t + n → R t , ( y , x ) y isproper.A point η ∈ R t is a non-proper point of the restriction of π to V if and only π − ( U ) ∩ V ∩ R t + n is not compact forany compact neighborhood U of η in R t . Example 3.
We consider the ideal h x + y − i . One can easily see that x is in Noether position with respect to thisideal as the equation x + y − is monic in x .On the other hand, the variable x is not in Noether position with respect to the ideal h xy − i . This can be observedgeometrically as the fiber at y = 0 of the projection of V ( xy − to the y -space lies in infinity.Another example is the ideal h yx + 2 x − i . The variable x is not in Noether position with respect to this ideal. Thefiber at y = 0 of the projection of V ( yx + 2 x − to the y -space contains a point (1 / , and a point at infinity. So,this projection is not proper. Gröbner bases and zero-dimensional ideals
Let F be a field and F be its algebraic closure. We denote by F [ x ] thepolynomial algebra in the variables x = ( x , . . . , x n ) . We fix an admissible monomial ordering ≻ (see Section 2.2,Cox et al. (2007)) over F [ x ] . For a polynomial p ∈ F [ x ] , the leading monomial of p with respect to ≻ is denoted by lm ≻ ( p ) .Given an ideal I ⊂ F [ x ] , the initial ideal of I with respect to the ordering ≻ is the ideal h lm ≻ ( p ) | p ∈ I i . AGröbner basis G of I with respect to the ordering ≻ is a generating set of I such that the set of leading monomials { lm ≻ ( g ) | g ∈ G } generates the initial ideal h lm ≻ ( p ) | p ∈ I i .For any polynomial p ∈ F [ x ] , the remainder of the division of p by G using the monomial ordering ≻ is uniquelydefined. It is called the normal form of p with respect to G and is denoted by NF G ( p ) . A polynomial p is reduced by G if p coincides with its normal form in G . A Gröbner basis G is said to be reduced if, for any g ∈ G , all terms of g are reduced modulo the leading terms of G .An ideal I is said to be zero-dimensional if the algebraic set V ( I ) ⊂ F n is finite and non-empty. By (Cox et al.,2007, Sec. 5.3, Theorem 6), the quotient ring F [ x ] /I is a F -vector space of finite dimension. The dimension ofthis vector space is also called the algebraic degree of I ; it coincides with the number of points of V ( I ) countedwith multiplicities (Basu et al., 2006, Sec. 4.5). For any Gröbner basis of I , the set of monomials in F [ x ] which areirreducible by G forms a monomial basis, which we call B , of this vector space. For any p ∈ F [ x ] , the normal form of p by G can be interpreted as the image of p in F [ x ] /I and is a linear combination of elements of B (with coefficientsin F ). Therefore, the operations in the quotient algebra F [ x ] /I such as vector additions or scalar multiplications canbe computed explicitly using the normal form reduction.In this article, while working with polynomial systems depending on parameters in Q [ y ][ x ] , we frequently take F tobe the rational function field Q ( y ) and treat polynomials in Q [ y ][ x ] as elements of Q ( y )[ x ] . Zero-dimensional parametrizations A zero-dimensional parametrization R of coefficients in Q consists of a se-quence of polynomials ( w, v , . . . , v n ) ∈ ( Q [ u ]) n +1 with a new variable u and ( a , . . . , a n ) ∈ Q n such that w issquare-free and u = ( P ni =1 a i · v i /w ′ ) mod w . The solution set of R , defined as (cid:26)(cid:18) v ( ϑ ) w ′ ( ϑ ) , . . . , v n ( ϑ ) w ′ ( ϑ ) (cid:19) ∈ C n | ϑ ∈ C such that w ( ϑ ) = 0 (cid:27) , is finite and is denoted by Z ( R ) .A finite algebraic set V ∈ C n is said to be represented by a zero-dimensional parametrization R if and only if V coincides with Z ( R ) .Note that it is possible to retrieve a polynomial parametrization by inverting the derivative w ′ modulo w . Still, thisrational parametrization whose denominator is the derivative of w is known to be better for practical computations asit usually involves coefficients with smaller bit size (see Dahan and Schost (2004)). Rational parametrizations
We consider now a parametric system f ∈ Q [ y ][ x ] where y = ( y , . . . , y t ) are param-eters and x = ( x , . . . , x n ) are variables. Under some extra assumptions on the system f , (Schost, 2003, Theorem1) proves the existence of a sequence ( w, v , . . . , v n ) ⊂ ( Q ( y )[ u ]) n with a new variable u and a Zariski open subset Y ⊂ C t such that• w is a square-free polynomial in Q ( y )[ u ] .• u = P ni =1 a i x i for some ( a , . . . , a n ) ∈ Q n . 8 For η ∈ Y , η does not cancel any denominator of ( w, v , . . . , v n ) and V ( f ( η, · )) = (cid:26)(cid:18) v ∂w/∂u ( η, ϑ ) , . . . , v n ∂w/∂u ( η, ϑ ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) w ( η, ϑ ) = 0 , ∂w∂u ( η, ϑ ) = 0 (cid:27) . The sequence ( w, v , . . . , v n ) is called a rational parametrization of f . It can be computed using the parametricgeometric resolution algorithm which is described in Schost (2003).Intuitively, this parametrization provides a generic description for the solutions of f ( η, · ) when η ranges over C t . Itgeneralizes the notion of zero-dimensional parametrizations to the parametric setting. In this section, we study the following algorithmic problem. Given ( g , . . . , g s ) in Q [ y , . . . , y t ] , compute at least onesample point per connected component of the semi-algebraic set S ⊂ R t defined by g = 0 , . . . , g s = 0 . Such sample points will be encoded with zero-dimensional parametrizations which we described in Section 2.The main result of this section which will be used in the sequel of this paper is the following.
Theorem III.
Let ( g , . . . , g s ) in Q [ y , . . . , y t ] with D ≥ max ≤ i ≤ s deg( g i ) and S ⊂ R t be the semi-algebraic setdefined by g = 0 , . . . , g s = 0 . There exists a probabilistic algorithm which on input ( g , . . . , g s ) outputs a finite family of zero-dimensionalparametrizations R , . . . , R k encodes at most (2 sD ) t points such that ∪ ki =1 Z ( R i ) meets every connected compo-nent of S using O e(cid:18)(cid:18) D + tt (cid:19) (2 t ) s t +1 t D t +1 (cid:19) . arithmetic operations in Q . The rest of this section is devoted to the proof of this theorem.
Proof.
By (Faugère et al., 2008, Lemma 1), there exists a non-empty Zariski open set
A × E ⊂ C s × C such thatfor ( a = ( a , . . . , a s ) , e ) ∈ A × E ∩ R s × R , the following holds. For I = { i , . . . , i ℓ } ⊂ { , . . . , s } and σ =( σ , . . . , σ s ) ∈ {− , } s , the algebraic sets V I ,σ a ,e ⊂ C t defined by g i + σ i a i e = · · · = g i ℓ + σ i ℓ a i ℓ e = 0 are, either empty, or ( t − ℓ ) -equidimensional and smooth, and the ideal generated by their defining equations is radical.Note that by the transfer principle, one can choose instead of a scalar e an infinitesimal ε so that the algebraic sets V I ,σ a ,ε and their defining set of equations satisfy the above properties. When, in the above equations, one leaves ε asa variable, one obtains equations defining an algebraic set in C t +1 . We denote by V I ,σ a ,ε the union of the ( t + 1 − ℓ ) -equidimensional components of this algebraic set.Further we also assume that the a i ’s are chosen positive.Denote by S ( ε ) the extension of the semi-algebraic set S to R h ε i t ; similarly, the extension of any connected component C of S to R h ε i t is denoted by C ( ε ) .Now, remark that any connected component C ( ε ) of S ( ε ) contains a connected component of the semi-algebraic set S ( ε ) a defined by: ( − a ε ≥ g ∨ g ≥ a ε ) ∧ · · · ∧ ( − a s ε ≥ g s ∨ g s ≥ a s ε ) Hence, we are led to compute sample points per connected component of S ( ε ) a . These will be encoded with zero-dimensional parametrizations with coefficients in Q [ ε ] .By (Basu et al., 2006, Proposition 13.1), in order to compute sample points per connected component in S ( ε ) a , itsuffices to compute sample points in the real algebraic sets V I ,σ a ,ε ∩ R t . To do that, since the algebraic sets V I ,σ a ,ε satisfy9he above regularity properties, we can use the algorithm and geometric results of Safey El Din and Schost (2003). Tostate these results, one needs to introduce some notation.Let Q be a real field, R be a real closure of Q and C be an algebraic closure of R . For an algebraic set V ⊂ C t defined by h = · · · = h ℓ = 0 ( h i ∈ Q [ y ] with y = ( y , . . . , y t ) ) and M ∈ GL t ( R ) , we denote by V M the set { M − · x | x ∈ V } and, for ≤ i ≤ ℓ , by h iM the polynomial h i ( M · y ) and by π i the canonical projection ( y , . . . , y t ) → ( y , . . . , y i ) ( π will simply denote ( y , . . . , y t ) → {•} ). By slightly abusing notation, we will alsodenote by π i projections from V I ,σ a ,ε to the first i coordinates ( y , . . . , y i ) .We will consider the set of critical points of the restriction of π i to V and will denote this set by crit( π i , V ) for ≤ i ≤ ℓ . By (Safey El Din and Schost, 2003, Theorem 2), for a generic choice of M ∈ GL t ( R ) , the union of V M ∩ π − t − ℓ (0) with the sets crit ( π i , V M ) ∩ π − i − (0) (for ≤ i ≤ t − ℓ ) is finite and meets all connected componentsof V M ∩ R t . Because V satisfies the aforementioned regularity assumptions, crit ( π i , V M ) ∩ π − i − (0) is defined as theprojection on the y -space of the solution set to the polynomials h M , ( λ , . . . , λ ℓ ) .jac ( h M , i ) , u λ + · · · + u ℓ λ ℓ = 1 , y = · · · = y i − = 0 , where h = ( h , . . . , h ℓ ) , λ , . . . , λ ℓ are new variables (called Lagrange multipliers), jac ( h M , i ) is the Jacobianmatrix associated to h M truncated by forgetting its first first i columns and the u i ’s are generically chosen (see also(Safey El Din and Schost, 2017, App. B)).Recall that D denotes the maximum degree of the h j ’s and let E be the length of a straight-line program evaluating h . Observe now that, setting the y j ’s to (for ≤ j ≤ i − ), and using (Safey El Din and Schost, 2018, Theorem1) combined with the degree estimates in (Safey El Din and Schost, 2018, Section 5), we obtain that such systems canbe solved using O (cid:18)(cid:18) t − iℓ (cid:19) D ℓ ( D − t − ( i − − ℓ (cid:19) ( E + ( t + ℓ ) D + ( t + ℓ ) )( t + ℓ ) ! arithmetic operations in Q and have at most (cid:18) t − iℓ (cid:19) D ℓ ( D − t − ( i − − ℓ solutions.Going back to our initial problem, one then needs to solve polynomial systems which encode the set crit( π i , V I ,σ a ,ε ) ofcritical points of the restriction of π i to V I ,σ a ,ε . Note that these systems have coefficients in Q [ ε ] . To solve such systems,we rely on Schost (2003), which consists in specializing ε to a generic value e ∈ Q and compute a zero-dimensionalparametrization of the solution set to the obtained system (within the above arithmetic complexity over Q ) and nextuse Hensel lifting and rational reconstruction to deduce from this parametrization a zero-dimensional parametrizationwith coefficients in Q ( ε ) . By (Schost, 2003, Corollary 1) and multi-homogeneous bounds on the degree of the criticalpoints of π i to V I ,σ a ,ε as in (Safey El Din and Schost, 2018, Section 5), this lifting step has a cost O e (( t + ℓ ) + ( t + ℓ + 1) E ) (cid:18)(cid:18) t − iℓ (cid:19) D ℓ ( D − t − ( i − − ℓ (cid:19) ! Hence, all in all computing one zero-dimensional parametrization for one critical locus uses O e (( t + ℓ ) D + ( t + ℓ + 1) E ) (cid:18)(cid:18) t − iℓ (cid:19) D ℓ ( D − t − ( i − − ℓ (cid:19) ! arithmetic operations in Q . Note that following Schost (2003), the degrees in ε of the numerators and denominators ofthe coefficients of these parametrizations are bounded by (cid:0) tℓ (cid:1) D ℓ ( D − t − ℓ .Summing up for all critical loci and using t − ℓ X i =0 (cid:18) t − iℓ (cid:19) = (cid:18) t + 1 ℓ + 1 (cid:19) we need to compute for a fixed V I ,σ a ,ε uses O e (( t + ℓ ) D + ( t + ℓ + 1) E ) (cid:18) t + 1 ℓ + 1 (cid:19) (cid:0) D ℓ ( D − t − ℓ (cid:1) ! Q . Also, the number of points computed this way is dominated by (cid:18) t + 1 ℓ + 1 (cid:19) (cid:0) D ℓ ( D − t − ℓ (cid:1) . Taking the sum for all possible algebraic sets V I ,σ a ,ε and remarking that• the sum of number of indices of cardinality ℓ for ≤ ℓ ≤ t is bounded by s t ;• the number of sets σ for a given ℓ is bounded by t ;• the sum P tℓ =0 (cid:0) t +1 ℓ +1 (cid:1) equals (cid:0) t +1 t (cid:1) − one deduces that all these zero-dimensional parametrizations can be computed within O e(cid:18) s t t (cid:18) t + 1 t (cid:19) (cid:0) (2 t ) D + (2 t + 1)Γ (cid:1) D t (cid:19) arithmetic operations in Q (recall that Γ bounds the length of a straight line program evaluating all the polynomialsdefining our semi-algebraic set S ) which we simplify to O e(cid:0) Γ (2 t ) s t t D t +1 (cid:1) . Similarly, using the above simplifications, the total number of points encoded by these zero-dimensional parametriza-tions is bounded above by (2 sD ) t .At this stage, we have just obtained zero-dimensional parametrizations with coefficients in Q ( ε ) .The above bound on the number of returned points is done but it remains to show how to specialize ε in order to getsample points per connected components in S . To do that, given a parametrization R ε = ( w, v , . . . , v t ) ⊂ Q ( ε )[ u ] t +1 ,we need to find a specialization value e for ε to obtain a parametrization R e such that• the number of real roots of the zero set associated to R e is the same as the number of real roots of the zeroset associated to R ε ;• when η ranges over the interval ]0 , e ] the signs of the g i ’s at the zero set associated to η does not vary.To do that, it suffices to choose e such that it is smaller than the smallest positive root of the resultant associated to (cid:0) w, ∂w∂u (cid:1) and the smallest positive roots of the resultant associated to w and g i (cid:16) v ∂w/∂u , . . . , v t ∂w/∂u (cid:17) . The algebraiccost (i.e. the resultant computations) are dominated by the complexity estimates of the previous step.Finally, note that Γ can be bounded by s (cid:0) D + tt (cid:1) when the g i ’s are given in an expanded form in the monomialbasis. Therefore, the arithmetic complexity for computing sample points of the semi-algebraic set defined by g = 0 , . . . , g s = 0 can be bounded by O e(cid:18)(cid:18) D + tt (cid:19) (2 t ) s t +1 t D t +1 (cid:19) . We end this section with a Corollary which is a consequence of the proof of (Basu et al., 2006, Theorem 13.18).Basically, once we have the parametrizations computed by the algorithm on which Theorem III relies, one can computesample points per connected components of the semi-algebraic set S within the same arithmetic complexity bounds.The idea is just to evaluate the g i ’s at these rational parametrizations and use bounds on the minimal distance betweentwo roots of a univariate polynomial such as (Basu et al., 2006, Prop. 10.22). Hence, the proof of the corollary belowfollows mutatis mutandis the same steps as the one of (Basu et al., 2006, Theorem 13.18). Corollary 4.
Let ( g , . . . , g s ) in Q [ y , . . . , y t ] with D ≥ max ≤ i ≤ s deg( g i ) and S ⊂ R t be the semi-algebraic setdefined by g = 0 , . . . , g s = 0 . There exists a probabilistic algorithm which on input ( g , . . . , g s ) outputs a finite set of points P in Q t of cardinalityat most (2 sD ) t points such that P meets every connected component of S using O e(cid:18)(cid:18) D + tt (cid:19) (2 t ) s t +1 t D t +1 (cid:19) . arithmetic operations in Q . Sturm based classical algorithm
In this section, we describe an algorithm based on Sturm’s theorem for solving Problem (1) and discuss its shortcom-ings.We consider a sequence f = ( f , . . . , f m ) ⊂ Q [ y ][ x ] where y = ( y , . . . , y t ) and x = ( x , . . . , x n ) . Let d be anupper bound of the total degree of the f i ’s. We require that the input system f satisfies the properties below. Assumption B . Let f be the above parametric polynomial system and V be the algebraic set defined by f . We saythat f satisfies Assumptions ( B ) if the following properties hold. (B1) The ideal generated by f is radical. (B2) The algebraic set V is equi-dimensional of dimension t . (B3) The restriction of π : ( y , x ) y to V is dominant. It is well-known that the above assumptions are satisfied by sufficiently generic systems (see e.g.Safey El Din and Schost (2018)).In what follows, we rely on the existence of a parametric geometric resolution (Schost, 2003) to reduce our initialmultivariate problem to a univariate one.Using (Schost, 2003, Proposition 2) with Assumption ( B1 ), there exists a non-empty open Zariski set A of C n suchthat, for ( a , . . . , a n ) ∈ Q n ∩ A , there exists a parametric geometric resolution ( w a , v , . . . , v n ) ⊂ ( Q ( y )[ x ]) n of f that satisfies the following properties.• w a is a square-free polynomial in Q [ y ][ x ] .• u = P ni =1 a i x i .• There exists a non-empty Zariski open subset Y a ⊂ C t such that, for η ∈ Y a , we have that V ( f ( η, · )) = (cid:26)(cid:18) v ∂w a /∂u ( η, ϑ ) , . . . , v n ∂w a /∂u ( η, ϑ ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) w a ( η, ϑ ) = 0 , ∂w a ∂u ( η, ϑ ) = 0 (cid:27) . The set Y a can be chosen as the set where the leading coefficient of w a , the resultant of w a and ∂w a /∂u ,and the denominators appearing in the coefficients of v , . . . , v n do not vanish.As a consequence, for η ∈ Y a , the number of complex solutions of f ( η, · ) is invariant and equals to the partial degreeof w a in u . We denote by D the partial degree of w a in u . By Bézout’s inequality (see e.g. Heintz (1983)), D isbounded above by d n .Let η ∈ C t and w a ( η, · ) be the specialization of the y variables in w a at η . From the existence of such a parametricresolution, we deduce that, for η ∈ Y a , the map ϕ : ( x , . . . , x n ) P ni =1 a i x i is a bijection between the complexroots of f ( η, · ) and w a ( η, · ) . Lemma 5.
Let f be a parametric system satisfying Assumption ( B ) and w a be the eliminating polynomial in theparametric geometric resolution of f as above. Then, we have V ( h f , . . . , f m , u − n X i =1 a i x i i ∩ Q [ y ][ u ]) = V ( w a ) . Consequently, the total degree of w a is at most d n .Proof. We prove that, under Assumption ( B ), there exists a square-free polynomial w ∈ Q [ y ][ x ] satisfying V ( h f , . . . , f m , u − n X i =1 a i x i i ∩ Q [ y ][ u ]) = V ( w ) . Let π u : C t + n +1 → C t +1 , ( y , x , u ) ( y , u ) and V u be the algebraic set defined by h f , u − P ni =1 a i x i i . Note that V and V u are isomorphic taking the map ( y , x ) ( y , x , P ni =1 a i x i ) as an isomorphism between them. Then, as thealgebraic set V satisfies Assumption ( B ), V u is equi-dimensional of dimension t and the restriction of Π : C t + n +1 → C t , ( y , x , u ) y to V u is dominant. Therefore, the Zariski closure of π u ( V u ) is an equi-dimensional algebraic setof dimension t . Hence, there exists a square-free polynomial w ∈ Q [ y ][ u ] such that V ( w ) = π u ( V u ) . Therefore, weobtain V ( h f , . . . , f m , u − P ni =1 a i x i i ∩ Q [ y ][ u ]) = π u ( V u ) = V ( w ) .12t remains to show that w a equals to w up to a constant. By the definition of parametric geometric resolution, for η ∈ Y a , then w a ( η, · ) and w ( η, · ) share the same complex roots. Therefore, w a equals to w up to a factor in Q [ y ] .However, both w a and w do not contain such kind of factor by Assumption ( B ).By Bézout’s inequalities, the degree of V ( f , . . . , f n , u − P ni =1 a i x i ) is at most d n . Hence, the degree of π u ( V u ) isalso bounded by d n . Therefore, the total degree of w a is bounded by d n .Recall that Y a is the non-empty Zariski open subset of C t where the leading coefficient of w a , the resultant of w a and ∂w a /∂u , and the denominators appearing in the coefficients of v , . . . , v n do not vanish. Lemma 6 shows that thenumbers of real roots of f ( η, · ) and w a ( η, · ) also coincide over Y a . Lemma 6.
Let Y a as above. Then, for η ∈ Y a ∩ R t , the numbers of real solutions of w a ( η, · ) and f ( η, · ) are equal.Proof. Let η ∈ R t ∩ Y a . By definition of Y a , the restriction of ϕ : ( x , . . . , x n ) P ni =1 a i x i to V ( f ( η, · )) isa bijection of between the complex roots of f ( η, · ) and w a ( η, · ) . As the sequence f ( η, · ) contains polynomials ofcoefficients in R , the non-real complex roots of f ( η, · ) appears as pairs of conjugate complex points of C n . Assumethat there exists a complex root whose image by ϕ is a real root of w a ( η, · ) , then its conjugate is also mapped to thesame real root. This contradicts the bijectivity of ϕ . Therefore, the numbers of real solutions of f ( η, · ) and w a ( η, · ) coincide.For h ∈ Q [ y ][ u ] of degree D in u , we denote by Σ (cid:18) h, ∂h∂u (cid:19) = ( s , . . . , s D ) ⊂ Q [ y ] the leading coeffients of the subresultant sequence associated to ( h, ∂h/∂u ) (see (Basu et al., 2006, Chap. 4)). Herewe enumerate this sequence in a way such that s is the leading coefficient of h as a polynomial in u and s D is theresultant of h and ∂h/∂u .We recall the specialization property of subresultant coefficients (see e.g. (Basu et al., 2006, Proposition 8.74)). For η ∈ R t that does not cancel the leading coefficient of h as a polynomial in u , then the subresultant coefficients of h ( η, · ) and ∂h ( η, · ) /∂u are exactly the evaluation of ( s , . . . , s D ) at η .By (Basu et al., 2006, Theorem 4.32), the number of real roots of h ( η, · ) equals the generalized permanences minusvariations (see (Basu et al., 2006, Notation 4.30)) of ( s , . . . , s D ) η . Note that this value is uniquely defined upon asign pattern of ( s , . . . , s D ) η .We can now describe Algorithm 1 which takes as input a sequence f = ( f , . . . , f m ) ⊂ Q [ y ][ x ] satisfying Assumption( B ) and it outputs semi-algebraic formulas solving Problem (1).It uses the following subroutines:• EliminatingPolynomial which takes as input f and outputs an eliminating polynomial w a , i.e., the firstpolynomial in a parametric geometric resolution of f .Such an algorithm can be derived from the probabilistic algorithm given in Schost (2003) that computesparametric geometric resolutions.• SubresultantCoefficients which takes as input w a and outputs Σ( w a , ∂w a /∂u ) = ( s , . . . , s D ) .We refer to (Basu et al., 2006, Algo. 8.77) for the explicit description of such an algorithm.• SamplePoints which takes as input the subresultant coefficients ( s , . . . , s D ) ⊂ Q [ y ] and outputs atleast one point per connected components of the semi-algebraic set defined by { s i = 0 | ≤ i ≤ D, s i is not a constant } .We refer to Section 3 for the explicit description of such an algorithm.• PermanencesMinusVariations which takes as input a sequence ( s , . . . , s D ) η and return its generalizedpermanences minus variations.Using (Basu et al., 2006, Notation 4.30), we can easily design such a subroutine.13 lgorithm 1: RRC-Sturm
Input:
A parametric system f ⊂ Q [ y ][ x ] satisfying Assumption ( B ) Output:
Semi-algebraic descriptions solving Problem (1) for the input f w a ← EliminatingPolynomial ( f ) ( s , . . . , s D ) ← SubresultantCoefficients ( w a , ∂w a /∂u, u ) L ← SamplePoints ( { s i = 0 | ≤ i ≤ D, s i is not a constant } ) for η ∈ L do r η ← PermanencesMinusVariations (( s , . . . , s D ) η ) end return { (sign ( s , . . . , s D ) η , η, r η ) | η ∈ L } Theorem I.
Let f = ( f , . . . , f n ) ⊂ Q [ y ][ x ] be a parametric system and d be the largest total degree among the deg( f i ) ’s. We assume that f satisfies Assumption ( B ) .Then, Algorithm 1, which is probabilistic, computes semi-algebraic formulas solving Problem 1 within O e(cid:18)(cid:18) t + 2 d n t (cid:19) t d nt +3 n (cid:19) arithmetic operations in Q . These semi-algebraic formulas contains polynomials in Q [ y ] of degree bounded by d n .Proof. We start with the correctness statement. Recall that s is the leading coefficient of w a as a polynomial in u .By (Basu et al., 2006, Proposition 8.74), for η ∈ C t that does not cancel s , the subresultant coefficients of w a ( η, · ) and ∂w a ( η, · ) /∂u is the specialization of ( s , . . . , s D ) at η . Therefore, from (Basu et al., 2006, Theorem 4.33), thenumber of real roots of w a ( η, · ) can be derived from the sign of the sequence ( s , . . . , s D ) η for η V ( s ) .On the other hand, the semi-algebraic set S defined by { s i = 0 | ≤ i ≤ D, s i is not a constant } is composed ofopen semi-algebraic connected components, namely S , . . . , S ℓ . Over each of them, the subresultant coefficients s i aresign-invariant. Thus, the number of distinct real roots of w a ( η, · ) is invariant when η varies in S i for each ≤ i ≤ ℓ .Recall that Y a ⊂ C t is the non-empty Zariski open set in Lemma 6 such that for η ∈ Y a , the numbers of real rootsof f ( η, · ) and w a ( η, · ) coincide. Therefore, the number of real solutions of f ( η, · ) is also invariant when η varies in S i ∩ Y a .Let L be the set of sample points of S . We deduce from the above arguments that the semi-algebraic sets defined by ∧ Di =1 sign ( s i ) = sign ( s i ( η )) for η ∈ L satisfy the requirement of Problem 1. The correctness of our algorithm isthen proven.By Lemma 5, the total degree of w a is bounded above by d n . Using (Basu et al., 2006, Proposition 8.71) on thepolynomial w a and ∂w a /∂u , we obtain the bound deg s j ≤ d n (2 j − ≤ d n for ≤ j ≤ D .We are now able to analyze the complexity of Algorithm 1.By (Schost, 2003, Corollary 1), running EliminatingPolynomial on input f = ( f , . . . , f n ) where the total degree ofeach f i is bounded by d takes O e(cid:18)(cid:18) d n + tt (cid:19) d n (cid:19) arithmetic operations in Q .The subresultant coefficients of w a and ∂w a /∂u can be computed using an evaluation-interpolation scheme as fol-lows. As the degree of s i is bounded by d n , we need to compute the subresultant coefficients of the evaluation of ( w a , ∂w a /∂u ) at (cid:0) t +2 d n t (cid:1) distinct points. Note that (cid:0) t +2 d n t (cid:1) is bounded by t d nt .Using (Basu et al., 2006, Algo. 8.77), it yields an arithmetic complexity O (cid:0) d n (cid:1) for each of those subresultantcomputations. Hence, in total, the specialized subresultant coefficients can be computed by O (cid:0) t d nt +2 n (cid:1) .Next, the cost of interpolating the s i ’s can be bounded by O e(cid:0) D t d nt log (cid:0) t d nt (cid:1)(cid:1) using the interpolation given inCanny et al. (1989). Thus, the arithmetic complexity of SubresultantCoefficients lies in O e(cid:0) D t d nt log (cid:0) t d nt (cid:1)(cid:1) . We rely on Corollary 4 for estimating the complexity of
SamplePoints . Using the algorithm of Section 3 (seeTheorem III and Corollary 4) on the sequence ( s , . . . , s D ) , one can compute sample points per connected components14f the semi-algebraic set defined by { s i = 0 | ≤ i ≤ D, s i is not a constant } in time O e(cid:18)(cid:18) t + 2 d n t (cid:19) t d nt + n t (cid:0) d n (cid:1) t +1 (cid:19) ≃ O e(cid:18)(cid:18) t + 2 d n t (cid:19) t d nt +3 n (cid:19) . By Corollary 4, this subroutine outputs a finite subset of Q t whose cardinal is bounded by t d nt . Using (Basu et al.,2006, Algorithm 9.4) to compute the permanences minus variations leads to a complexity of O (cid:0) t d nt + n (cid:1) .Summing up all the partial costs, we conclude that Algorithm 1 runs within O e(cid:18)(cid:18) t + 2 d n t (cid:19) t d nt +3 n (cid:19) arithmetic operations in Q . Example 7.
We will illustrate the algorithms of this paper using the sequence f = ( x + x − y , x x + y x + y x ) . We choose u = x when running Algorithm 1 (in a reasonable implementation, one would pick randomly a linear formbut we choose this one to obtain smaller data).We obtain the following rational parametrization ( w, v , v ) with w = u + 2 y u + ( y + y − y ) u − y y u − y y ,v = 2 y u + (cid:0) y + 2 y − y (cid:1) u − y y u − y y ,v = 2 y u + 2 y y y . The subresultant coefficients associated to (cid:0) w, ∂w∂u (cid:1) are: s = 1 , s = 1 , s = − y + y + 2 y ,s = − y − y y − y y + 3 y y − y y y + y y − y y − y y + y ,s = ( y y ) y ( − y − y y − y y − y + 3 y y − y y y + 3 y y − y y − y y + y ) . Since s and s are constants, we then compute at least one point per connected component of the semi-algebraic setdefined by s = 0 ∧ s = 0 ∧ s = 0 . This is done using e.g. the
RAGlib (Real Algebraic Geometry library) (Safey El Din, 2017). We obtain points andfind that the realizable sign conditions for ( s , s , s ) are [ − , − , − , [ − , − , , [ − , , , [1 , − , − , [1 , − , , [1 , , − , [1 , , . Applying (Basu et al., 2006, Theorem 4.32), we deduce the corresponding numbers of real roots to these sign patterns real root → ( s < ∧ s < ∧ s > ∨ ( s < ∧ s > ∧ s > ∨ ( s > ∧ s < ∧ s > real roots → ( s < ∧ s < ∧ s < ∨ ( s > ∧ s < ∧ s < ∨ ( s > ∧ s > ∧ s < real roots → s > ∧ s > ∧ s > Note that the maximum degree of the polynomials involved in the above formulas is . By contrast, observe thatthe restriction of the projection π : ( y , x ) → y to the real algebraic set defined by f is proper. Hence, applying asemi-algebraic variant of Thom’s isotopy lemma as in (Bonnard et al., 2016), one can deduce that the set of criticalvalues of this map discriminates the regions of the parameters’ space over which the number of real roots of f remainsinvariant.Using immediate Gröbner bases computations, one obtains that the Zariski closure of this set of critical values isdefined by the vanishing of y ( − y − y y − y y − y + 3 y y − y y y + 3 y y − y y − y y + y ) which has only degree . In this section, we adapt the construction encoding Hermite’s quadratic forms, also known as Hermite matrices to thecontext of parametric systems and describe an algorithm for computing those parametric Hermite matrices .15 .1 Definition
Let K be a field and I ⊂ K [ x ] be a zero-dimensional ideal. Recall that the quotient ring A K = K [ x ] /I is a K -vectorspace of finite dimension (Cox et al., 2007, Section 5.3, Theorem 6). The multiplication maps of A K are defined asfollows. Definition 8.
For any p ∈ K [ x ] , the multiplication map L p is defined as L p : A K → A K ,q p · q, where q and p · q are respectively the classes of q and p · q in the quotient ring A K . Note that the map L p is an endomorphism of A K as a K -vector space. The Hermite quadratic form associated to I isdefined as the bilinear form that sends ( p, q ) ∈ A K × A K to the trace of L p · q as an endomorphism of A K .We refer to (Basu et al., 2006, Chap. 4) for more details about Hermite quadratic forms.Now, let f = ( f , . . . , f m ) be a polynomial sequence in Q [ y ][ x ] . We take the rational function field Q ( y ) as the basefield K and denote by h f i K the ideal generated by f in K [ x ] . We require that the system f satisfies Assumption ( A ).This leads to the following well-known lemma, which is the foundation for the construction of our parametric Hermitematrices. Lemma 9.
Assume that f satisfies Assumption ( A ) . Then the ideal h f i K is zero-dimensional.Proof. Assume that there exists a coordinate x i for ≤ i ≤ n such that h f i ∩ C [ y , x i ] = h i . We denote respectivelyby π i and ˜ π i the projections ( y , x ) ( y , x i ) and ( y , x i ) y . By the assumption above, π i ( V ) is the whole space C t +1 . Then, we have the identity C t +1 = ˜ π i − ( O ) ∪ ˜ π i − ( C t \ O ) , where O be the open Zariski subset of C t required in Assumption ( A ).As Assumption ( A ) holds, dim ˜ π i − ( O ) = t . Since ˜ π i is a map from C t +1 to C t , its fibers are of at most dimension . Therefore, we have that ˜ π i − ( C t \ O ) ≤ C t \ O ) ≤ t . This contradicts to the above identity above. Weconclude that, for ≤ i ≤ n , h f i ∩ C [ y , x i ] = h i .On the other hand, by Assumption ( A ), the Zariski-closure of π ( V ) is the whole parameter space C t . Thus, we havethat h f i ∩ C [ y ] = h i . Since h f i ∩ C [ y ] = ( h f i ∩ C [ y , x i ]) ∩ C [ y ] for every ≤ i ≤ n , there exists a polynomial p i ∈ h f i ∩ C [ y , x i ] whose degree with respect to x i is non-zero. Clearly, p i is an element of the ideal h f i K . Thus,there exists d i such that x d i i is a leading term in h f i K . Hence, h f i K is a zero-dimensional ideal.Lemma 9 allows us to apply the construction of Hermite matrices described in (Basu et al., 2006, Chap. 4) to paramet-ric systems as follows.Since the ideal h f i K is zero-dimensional by Lemma 9, its associated quotient ring A K = K [ x ] / h f i K is a finitedimensional K -vector space. Let δ denote the dimension of A K as a K -vector space.We consider a basis B = { b , . . . , b δ } of A K , where the b i ’s are taken as monomials in the variables x . Such a basiscan be derived from Gröbner bases as follows. We fix an admissible monomial ordering ≻ over the set of monomials inthe variables x and compute a Gröbner basis G with respect to the ordering ≻ of the ideal h f i K . Then, the monomialsthat are not divisible by any leading monomial of elements of G form a basis of A K .Recall that, for an element p ∈ K [ x ] , we denote by p the class of p in the quotient ring A K . A representative of p can be derived by computing the normal form of p by the Gröbner basis G , which results in a linear combination ofelements of B with coefficients in Q ( y ) .Assume now the basis B of A K is fixed. For any p ∈ K [ x ] , the multiplication map L p is an endomorphism of A K .Therefore, it admits a matrix representation with respect to B , whose entries are elements in Q ( y ) . The trace of L p can be computed as the trace of the matrix representing it. Similarly, the Hermite’s quadratic form of the ideal h f i K can be represented by a matrix with respect to B . This leads to the following definition. Definition 10.
Given a parametric polynomial system f = ( f , . . . , f m ) ⊂ Q [ y ][ x ] satisfying Assumption ( A ) . We fixa basis B = { b , . . . , b δ } of the vector space K [ x ] / h f i K . The parametric Hermite matrix associated to f with respectto the basis B is defined as the symmetric matrix H = ( h i,j ) ≤ i,j ≤ δ where h i,j = trace( L b i · b j ) .
16t is important to note that the definition of parametric Hermite matrices depends both on the input system f and thechoice of the monomial basis B . Example 11.
We consider the same system f = ( x + x − y , x x + y x + y x ) as in Example 7. The parametricHermite matrix associated to f with respect to the basis B = { , x , x , x } is − y − y − y + y + y ) ∗ − y + y + y ) 4 y y y y − y ) ∗ ∗ y − y + y ) 2( y − y y − y y ) ∗ ∗ ∗ y − y y + 2 y − y y + 2 y . Whereas, using the lexicographical ordering x ≻ x , we obtain the basis B = { , x , x , x } . The matrix associatedto f with respect to B is the following Hankel matrix: − y − y + 2 y + 2 y y y − y ∗ ∗ ∗ y − y y + 2 y − y y + 2 y ∗ ∗ ∗ − y y + 20 y y − y + 10 y y y ∗ ∗ ∗ − y + 30 y y − y y + 2 y + 6 y y − y y y − y y + 2 y . We remark that all the entries of the matrices above lie in Q [ y ] and that the entries of the second matrix are of higherdegree than the first one’s. In the previous subsection, we have defined parametric Hermite matrices assuming one knows a Gröbner basis G withrespect to some monomial ordering of the ideal h f i K where K = Q ( y ) and h f i K is the ideal of K [ x ] generated by f .Computing such a Gröbner basis may be costly as this would require to perform arithmetic operations over the field Q ( y ) (or Z /p Z ( y ) where p is a prime when tackling this computational task through modular computations). In thisparagraph, we show that one can obtain parametric Hermite matrices by considering some Gröbner bases of the ideal h f i ⊂ Q [ y , x ] (hence, enabling the use of efficient implementations of Gröbner bases such as the F /F algorithms(Faugere, 1999; Faugère, 2002)).Since the graded reverse lexicographical ordering ( grevlex for short) is known for yielding Gröbner bases of relativelysmall degree comparing to other orders, we prefer using this ordering to construct our parametric Hermite matrices.Further, we will use the notation grevlex ( x ) for the grevlex ordering among the variables x (with x ≻ · · · ≻ x n )and grevlex ( x ) ≻ grevlex ( y ) (with y ≻ · · · ≻ y t ) for the elimination ordering. We denote respectively by lm x ( p ) and lc x ( p ) the leading monomial and the leading coefficient of p ∈ K [ x ] with respect to the ordering grevlex ( x ) . Lemma 12.
Let G be the reduced Gröbner basis of h f i with respect to the elimination ordering grevlex ( x ) ≻ grevlex ( y ) . Then G is also a Gröbner basis of h f i K with respect to the ordering grevlex ( x ) .Proof. Since G is a Gröbner basis of the ideal h f i , every polynomial f i of f can be written as f i = P g ∈G c g · g where c g ∈ Q [ x , y ] . Therefore, any element of h f i K can also be written as a combination of elements of G with coefficientsin Q ( y )[ x ] . In other words, G is a set of generators of h f i K .Let p be a polynomial in K [ x ] , p is contained h f i K if and only if there exists a polynomial q ∈ Q [ y ] such that q · p ∈ h f i . Thus, the leading monomial of p as an element of K [ x ] with respect to the grevlex ordering grevlex ( x ) is contained in the ideal h lm x ( g ) | g ∈ Gi . Therefore, G is a Gröbner basis of h f i K .Hereafter, we denote by G the reduced Gröbner basis of h f i with respect to the elimination ordering grevlex ( x ) ≻ grevlex ( y ) . Let B be the set of all monomials in x that are not reducible by G , which is finite by Lemmas 9 and 12.The set B actually forms a basis of the K -vector space K [ x ] / h f i K . Then, we denote by H the parametric Hermitematrix associated to f with respect to this basis B .We consider the following assumption on the input system f . Assumption C . For g ∈ G , the leading coefficient lc x ( g ) does not depend on the parameters y . As the computations in the quotient ring A K are done through normal form reductions by G , the lemma below isstraight-forward. Lemma 13.
Under Assumption ( C ) , the entries of the parametric Hermite matrix H are elements of Q [ y ] . roof. Since Assumption ( C ) holds, the leading coefficients lc x ( g ) do not depend on parameters y for g ∈ G . Thenormal form reduction in A K of any polynomial in Q [ y ][ x ] returns a polynomial in Q [ y ][ x ] . Thus, each normal formcan be written as a linear combination of B whose coefficients lie in Q [ y ] . Hence, the multiplication map L b i · b j for ≤ i, j ≤ δ can be represented by polynomial matrices in Q [ y ] with respect to the basis B . As an immediateconsequence, the entries of H , as being the traces of those multiplication maps, are polynomials in Q [ y ] .The next proposition states that Assumption ( C ) is satisfied by a generic system f . It implies that the entries of theparametric Hermite matrix of a generic system with respect to the basis B derived from G completely lie in Q [ y ] . Wepostpone the proof of Proposition 14 to Subsection 7.1 where we prove a more general result (see Proposition 30). Proposition 14.
Let C [ x , y ] d be the set of polynomials in C [ x , y ] having total degree bounded by d . There exists anon-empty Zariski open subset F C of C [ x , y ] nd such that Assumption ( C ) is satisfied by any f ∈ F C ∩ Q [ x , y ] n . Recall that G is the reduced Gröbner basis of h f i with respect to the ordering grevlex ( x ) ≻ grevlex ( y ) and B is thebasis of K [ x ] / h f i K derived from G as discussed in the previous subsection. Then, H is the parametric Hermite matrixassociated to f with respect to the basis B .Let η ∈ C t and φ η : C ( y )[ x ] → C [ x ] , p ( y , x ) p ( η, x ) be the specialization map that evaluates the parameters y at η . Then f ( η, · ) = ( φ η ( f ) , . . . , φ η ( f m )) . We denote by H ( η ) the specialization ( φ η ( h i,j )) ≤ i,j ≤ δ of H at η .Recall that, for a polynomial p ∈ C ( y )[ x ] , the leading coefficient of p considered as a polynomial in the variables x with respect to the ordering grevlex ( x ) is denoted by lc x ( p ) . In this subsection, for p ∈ C [ x ] , we use lm( p ) to denotethe leading monomial of p with respect to the ordering grevlex ( x ) .Let W ∞ ⊂ C t denote the algebraic set ∪ g ∈G V (lc x ( g )) . In Proposition 16, we prove that, outside W ∞ , the specializa-tion H ( η ) coincides with the classic Hermite matrix of the zero-dimensional ideal f ( η, · ) ⊂ Q [ x ] . This is the mainresult of this subsection.Since the operations over the K -vector space A K rely on normal form reductions by the Gröbner basis G of h f i K ,the specialization property of H depends on the specialization property of G . Lemma 15 below, which is a directconsequence of (Kalkbrener, 1997, Theorem 3.1), provides the specialization property of G . We give here a moreelementary proof for this lemma than the one in Kalkbrener (1997). Lemma 15.
Let η ∈ C t \ W ∞ . Then the specialization G ( η, · ) := { φ η ( g ) | g ∈ G} is a Gröbner basis of the ideal h f ( η, · ) i ⊂ C [ x ] generated by f ( η, · ) with respect to the ordering grevlex ( x ) .Proof. Since η ∈ C t \ W ∞ , the leading coefficient lc x ( g ) does not vanish at η for every g ∈ G . Thus, lm x ( g ) =lm( φ η ( g )) .We denote by M the set of all monomials in the variables x and M G := { m ∈ M | ∃ g ∈ G : lm x ( g ) divides m } = { m ∈ M | ∃ g ∈ G : lm( φ η ( g )) divides m } . For any p ∈ h f i ⊂ Q [ x , y ] , we prove that lm( φ η ( f )) ∈ M G . If p is identically zero, there is nothing to prove. So,we assume that p = 0 , p is then expanded in the form below: p = X m ∈M G c m · m + X m ∈M\M G c m · m, where the c m ’s are elements of Q [ y ] . Since p is not identically zero, there exists m ∈ M G such that c m = 0 .Since G is a Gröbner basis of h f i K , any monomial in M G can be reduced by G to a unique normal form in K [ x ] .These divisions involve denominators, which are products of some powers of the leading coefficients of G with respectto the variables x . We write NF G ( p ) = X m ∈M G c m · NF G ( m ) + X m ∈M\M G c m · m. As p ∈ h f i K , we have that NF G ( p ) = 0 , which implies X m ∈M\M G c m · m = − X m ∈M G c m · NF G ( m ) . p = X m ∈M G c m · ( m − NF G ( m )) Since η does not cancel any denominator appearing in NF G ( m ) , we can specialize the identity above without anyproblem: φ η ( p ) = X m ∈M G φ η ( c m ) · ( m − φ η (NF G ( m ))) . If at least one of the φ η ( c m ) does not vanish, then the leading monomial of φ η ( f ) is in M G . Otherwise, if all the φ η ( c m ) are canceled, then φ η ( p ) is identically zero, and there is not any new leading monomial appearing either. So,the leading monomial of any p ∈ h f η i is contained in M G , which means G ( η, · ) is a Gröbner basis of h f ( η, · ) i withrespect to grevlex ( x ) . Proposition 16.
For any η ∈ C t \ W ∞ , the specialization H ( η ) coincides with the classic Hermite matrix of thezero-dimensional ideal h f ( η, · ) i ⊂ C [ x ] .Proof. As a consequence of Lemma 15, each computation in A K derives a corresponding one in C [ x ] / h f ( η, · ) i byevaluating y at η in every normal form reduction by G . This evaluation is allowed since η does not cancel anydenominator appearing during the computation. Therefore, we deduce immediately the specialization property of theHermite matrix.Using Proposition 16 and (Basu et al., 2006, Theorem 4.102), we obtain immediately the following corollary thatallows us to use parametric Hermite matrices to count the root of a specialization of a parametric system. Corollary 17.
Let η ∈ C t \ W ∞ , then the rank of H ( η ) is the number of distinct complex roots of f ( η, · ) . When η ∈ R t \ W ∞ , the signature of H ( η ) is the number of distinct real roots of f ( η, · ) .Proof. By Proposition 16, H ( η ) is a Hermite matrix of the zero-dimensional ideal h f ( η, · ) i . Then, (Basu et al., 2006,Theorem 4.102) implies that the rank (resp. the signature) of H ( η ) equals to the number of distinct complex (resp.real) solutions of f ( η, · ) .We finish this subsection by giving some explanation for what happens above W ∞ , where our parametric Hermitematrix H does not have good specialization property. Lemma 18.
Let W ∞ defined as above. Then W ∞ contains all the following sets: • The non-proper points of the restriction of π to V (see Section 2 for this definition). • The set of points η ∈ C t such that the fiber π − ( η ) ∩ V is infinite. • The image by π of the irreducible components of V whose dimensions are smaller than t .Proof. The claim for the set of non-properness of the restriction of π to V is already proven in (Lazard and Rouillier,2007, Theorem 2). We focus on the two remaining sets.Using the Hermite matrix, we know that for η ∈ C t \ W ∞ , the system f ( η, · ) admits a non-empty finite set of complexsolutions. On the other hand, for any η ∈ C t such that π − ( η ) ∩ V is infinite, f ( η, · ) has infinitely many complexsolutions. Therefore, the set of such points η is contained in W ∞ .Let V >t be the union of irreducible components of V of dimension greater than t . By the fiber dimension theorem(Shafarevich, 2013, Theorem 1.25), the fibers of the restriction of π to V >t must have dimension at least one. Similarly,the components of dimension t whose images by π are contained in a Zariski closed subset of C t also yield infinitefibers. Therefore, as proven above, all of these components are contained in π − ( W ∞ ) .We now consider the irreducible components of dimension smaller than t . Let V ≥ t and V 19e denote by R the polynomial ring Q ( y )[ x ] . Then, the above identity is transferred into R : I · R = ( I ≥ t · R ) ∩ ( I 6∈ W ∞ , we have that π − ( η ) ∩ V = π − ( η ) ∩ V ≥ t . This leads to π − ( C t \ W ∞ ) ∩ V ≥ t = π − ( C t \ W ∞ ) ∩ V . Then, π − ( C t \ W ∞ ) ∩ V DRL-Matrix Input: A parametric polynomial system f = ( f , . . . , f m ) Output: A parametric Hermite matrix H associated to f with respect to the basis B G , B ← GrobnerBasis ( f , grevlex ( x ) ≻ grevlex ( y )) G ′ ← ReduceGB ( G ) w ∞ ← sqfree (lcm g ∈G (lc x ( g ))) ( L x , . . . , L x n ) ← XMatrices ( G ′ , B ) ( L b , . . . , L b δ ) ← BMatrices (( L x , . . . , L x n ) , B ) H ← TraceComputing ( L b , . . . , L b δ ) return [ H , w ∞ ] Removing denominators Note that, through the computation in the quotient ring A K , the entries of our parametricHermite matrix possibly contains denominators that lie in Q [ y ] . As the algorithm that we introduce in Section 6 willrequire us to manipulate the parametric Hermite matrix that we compute, these denominators can be a bottleneckto handle the matrix. Therefore, we introduce an extra subroutine RemoveDenominator that returns a parametricHermite matrix H ′ of f without denominator. • RemoveDenominator that takes as input the matrix H computed by DRL-Matrix and outputs a matrix H ′ which is the parametric Hermite matrix associated to f with respect to a basis B ′ that will be made explicitbelow.As we can freely choose any basis of form { c i · b i | ≤ i ≤ δ } where the c i ’s are elements of Q [ y ] , weshould use a basis that leads to a denominator-free matrix. To do this, we choose c i as the denominator of trace( L b i ) (which lies in the first row of the matrix H computed by TraceComputing ). Then, for the entryof H that corresponds to b i and b j , we can multiply it with c i · c j . The output matrix H ′ is the parametricHermite matrix associated to f with respect to the basis { c i · b i | ≤ i ≤ δ } . It usually does not contain anydenominator and is handled easier in practice. Evaluation & interpolation scheme for generic systems Here we assume that the input system f satisfies Assump-tion ( C ). By Lemma 13, the entries of H are polynomials in Q [ y ] . Suppose that we know beforehand a value Λ that islarger than the degree of any entry of H , we can compute H by an evaluation & interpolation scheme as follows.We start by choosing randomly a set E of (cid:0) t +Λ t (cid:1) distinct points in Q t . Then, for each η ∈ E , we use DRL-Matrix (Al-gorithm 2) on the input f ( η, · ) to compute the classic Hermite matrix associated to f ( η, · ) with respect to the ordering grevlex ( x ) . These computations involve only polynomials in Q [ x ] and not in Q ( y )[ x ] . Finally, we interpolate theparametric Hermite matrix H from its specialized images H ( η ) computed previously.21ince Assumption ( C ) holds, then W ∞ is empty. By Proposition 16, the Hermite matrix of f ( η, · ) with respect to grevlex ( x ) is the image H ( η ) of H . Therefore, the above scheme computes correctly the parametric Hermite matrix H .We also remark that, in the computation of the specializations H ( η ) , we can replace the subroutine XMatrices in DRL-Matrix by a linear-algebra-based algorithm described in Faugère et al. (2013). That algorithm constructs the Macaulaymatrix and carries out matrix reductions to obtain simultaneously the normal forms that XMatrices requires.Assume a degree bound Λ is known, we estimate the arithmetic complexity for computing the parametric Hermitematrix in Proposition 19 below. We postpone to Subsection 7.1 for proving an explicit bound for Λ when the inputsystem satisfies some extra generic assumptions. Proposition 19. Assume that the system f = ( f , . . . , f m ) ⊂ Q [ y ][ x ] satisfying Assumptions ( A ) and ( C ) . Let δ bethe dimension of the K -vector space K [ x ] / h f i K where K = Q ( y ) . Let H be the parametric Hermite matrix associatedto f constructed using grevlex ( x ) ordering. Then, by Lemma 13, the entries of the parametric Hermite matrix H liein Q [ y ] .Let Λ be an upper degree bound of the entries of H . Using the evaluation & interpolation scheme, one can compute H within O e(cid:18)(cid:18) t + Λ t (cid:19) (cid:18) m (cid:18) d + n + tn + t (cid:19) + δ ω +1 + δ log (cid:18) t + Λ t (cid:19)(cid:19)(cid:19) arithmetic operations in Q , where, by Bézout’s bound, δ is bounded by d m .Proof. As the degrees of the entries of H are bounded by Λ . Following the evaluation & interpolation scheme requiresone to compute (cid:0) t +Λ t (cid:1) specialized Hermite matrices. We first analyze the complexity for computing each of thosespecialized Hermite matrices.The evaluation of f at each point η ∈ Q t costs O (cid:16) m (cid:0) d + n + tn + t (cid:1)(cid:17) arithmetic operations in Q .Next, we compute the matrices representing the L x i ’s using the linear algebra approach given in Faugère et al. (2013).It yields an arithmetic complexity of O ( nδ ω ) , where ω is the exponential constant for matrix multiplication.The traces of those matrices are then computed using nδ additions in Q . The subroutine BMatrices consists ofessentially δ multiplication of δ × δ matrices (with entries in Q ). This leads to an arithmetic complexity O ( δ ω +1 ) .Next, the computation of each entries h i,j is simply a vector multiplication of length δ , whose complexity is O ( δ ) .Thus, TraceComputing takes in overall O ( δ ) arithmetic operations in Q .Thus, every specialized Hermite matrix can be computed using O ( δ ω +1 ) arithmetic operations in Q . In total, thecomplexity of the evaluation step lies in O (cid:16)(cid:0) t +Λ t (cid:1) (cid:16) m (cid:0) d + n + tn + t (cid:1) + δ ω +1 (cid:17)(cid:17) .Finally, we interpolation δ entries which are polynomials in Q [ y ] of degree at most Λ . Using the multivariate inter-polation algorithm of Canny et al. (1989), the complexity of this step therefore lies in O e(cid:16) δ (cid:0) t +Λ t (cid:1) log (cid:0) t +Λ t (cid:1)(cid:17) .Summing up the both steps, we conclude that the parametric Hermite matrix H can be obtained within O e(cid:18)(cid:18) t + Λ t (cid:19) (cid:18) m (cid:18) d + n + tn + t (cid:19) + δ ω +1 + δ log (cid:18) t + Λ t (cid:19)(cid:19)(cid:19) arithmetic operations in Q . We present in this section two algorithms targeting the real root classification problem through parametric Hermitematrices. The one described in Subsection 6.1 aims to solve the weak version of Problem (1). The second algorithm,given in Subsection 6.2 outputs the semi-algebraic formulas of the cells S i that solves Problem (1). Further, in Section7, we will see that, for a generic sequence f , the semi-algebraic formulas computed by this algorithm consist ofpolynomials of degree bounded by n ( d − d n , which is better than the degree bound d n obtained by Algorithm 1and all previously known bounds.Throughout this section, our input is a parametric polynomial system f = ( f , . . . , f m ) ⊂ Q [ y ][ x ] . We require that f satisfies Assumptions ( A ) and that the ideal h f i is radical.22et G be the reduced Gröbner basis of the ideal h f i ⊂ Q [ x , y ] with respect to the ordering grevlex ( x ) ≻ grevlex ( y ) .Let K denote the rational function field Q ( y ) . We recall that B ⊂ Q [ x ] is the basis of K [ x ] / h f i K derived from G and H is the parametric Hermite matrix associated to f with respect to the basis B . (1)From Subsection 5.3, we know that, outside the algebraic set W ∞ := ∪ g ∈G V (lc x ( g )) , the parametric matrix H possesses good specialization property (see Proposition 16). We denote by w ∞ the square-free part of lcm g ∈G lc x ( g ) .This polynomial w ∞ is returned as an output of Algorithm 2. Note that V ( w ∞ ) = W ∞ . Lemma 20. When Assumption ( A ) holds and the ideal h f i is radical, the determinant of H is not identically zero.Proof. Recall that K denotes the rational function field Q ( y ) . We prove that the ideal h f i K ⊂ K [ x ] is radical.Let p ∈ K [ x ] such that there exists n ∈ N satisfying p n ∈ h f i K . Therefore, there exists a polynomial q ∈ Q [ y ] suchthat q · p n ∈ h f i . Then, ( q · p ) n ∈ h f i . As h f i is radical, we have that q · p ∈ h f i . Thus, p ∈ h f i K , which concludesthat h f i K is radical.By Lemma 9, h f i K is a radical zero-dimensional ideal in Q ( y ) . Since H is also a Hermite matrix (in the classic sense)of h f i K , H is full rank. Therefore, det( H ) is not identically zero.Let w H := n / gcd( n , w ∞ ) where n is the square-free part of the numerator of det( H ) . We denote by W H thevanishing set of w H . By Lemma 20, W H is a proper Zariski closed subset of C t . Our algorithm relies on thefollowing proposition. Proposition 21. Assume that Assumption ( A ) holds and the ideal h f i is radical. Then, for each connected component S of the semi-algebraic set R t \ ( W ∞ ∪ W H ) , the number of real solutions of f ( η, · ) is invariant when η varies over S .Proof. By Lemma 18, W ∞ contains the following sets:• The non-proper points of the restriction of π to V .• The point η ∈ C t such that the fiber π − ( η ) ∩ V is infinite.• The image by π of the irreducible components of V whose dimensions are smaller than t .Now we consider the set K ( π, V ) := sing( V ) ∪ crit( π, V ) . Let ∆ := jac( f , x ) be the Jacobian matrix of f withrespect to the variables x . The ideal generated by the n × n -minors of ∆ is denoted by I ∆ . Note that, since f isradical, K ( π, V ) is the algebraic set defined by the ideal h f i + I ∆ .By Proposition 16, for η ∈ C t \ W ∞ , h f i is a zero-dimensional ideal and the quotient ring C [ x ] / h f ( η, · ) i hasdimension δ . Moreover, if η ∈ C t \ ( W ∞ ∪ W H ) , the system f ( η, · ) has δ distinct complex solutions as the rank of H ( η ) is δ . Therefore, every complex root of f ( η, · ) is of multiplicity one (we use the definition of multiplicity givenin (Basu et al., 2006, Sec. 4.5)).Now we prove that, for such a point η , the fiber π − ( η ) does not intersect K ( π, V ) . Assume by contradiction thatthere exists a point ( η, χ ) ∈ C t + n lying in π − ( η ) ∩ K ( π, V ) . Note that χ is a solution of f ( η, · ) , i.e., f ( η, χ ) = 0 .As ( η, χ ) ∈ K ( π, V ) , then it is contained in V ( I ∆ ) . Hence, as the derivation in ∆ does not involve y , χ cancelsall the n × n -minors of the Jacobian matrix jac( f ( η, · ) , x ) . (Basu et al., 2006, Proposition 4.16) implies that χ hasmultiplicity greater than one. This contradicts to the claim that f ( η, · ) admits only complex solutions of multiplicityone.Therefore, we conclude that, for η ∈ C t \ ( W ∞ ∪ W H ) , π − ( η ) does not intersect K ( π, V ) .So, using what we prove above and Lemma 18, we deduce that, for η ∈ R t \ ( W ∞ ∪ W H ) , then there exists an openneighborhood O η of η for the Euclidean topology such that π − ( O η ) does not intersect K ( π, V ) ∪ π − ( W ∞ ) .Therefore, by Thom’s isotopy lemma (Coste and Shiota, 1992), the projection π realizes a locally trivial fibrationover R t \ ( W ∞ ∪ W H ) . So, for any connected component C of R t \ ( W ∞ ∪ W H ) and any η ∈ C , we have that π − ( C ) ∩ V ∩ R t + n is homeomorphic to C × ( π − ( η ) ∩ V ∩ R t + n ) .As a consequence, the number of distinct real solutions of f ( η, · ) is invariant when η varies over each connectedcomponent of R t \ ( W ∞ ∪ W H ) . 23o describe Algorithm 3, we need to introduce the following subroutines: • CleanFactors which takes as input a polynomial p ∈ Q [ y , x ] and the polynomial w ∞ . It computes thesquare-free part of p with all the common factors with w ∞ removed. • Signature which takes as input a symmetric matrix with entries in Q and evaluates its signature. • SamplePoints which takes as input a set of polynomials g , . . . , g s ∈ Q [ y ] and computes a finite subset R of Q t that intersects every connected component of the semi-algebraic set defined by ∧ si =1 g i = 0 . An explicitdescription of SamplePoints is given in the proof of Theorem III in Section 3.The pseudo-code of Algorithm 3 is below. Its proof of correctness follows immediately from Proposition 21 andCorollary 17. Algorithm 3: Weak-RRC-Hermite Input: A polynomial sequence f ∈ Q [ y ][ x ] such that h f i is radical and Assumptions ( A ) holds. Output: A set of sample points and the corresponding numbers of real solutions solving the weak version ofProblem (1) [ H , w ∞ ] ← DRL-Matrix ( f ) w H ← CleanFactors (numer(det( H )) , w ∞ ) L ← SamplePoints ( w H = 0 ∧ w ∞ = 0) for η ∈ L do r η ← Signature ( H ( η )) end return { ( η, r η ) | η ∈ L } Example 22. We continue with the system in Example 11. The determinant of its parametric Hermite matrix is w H = 16 y ( − y − y y − y y − y + 3 y y − y y y + 3 y y − y y − y y + y ) . We notice that w H coincides exactly with the output returned by the procedure D ISCRIMINANT V ARIETY of Maple’spackage R OOT F INDING [P ARAMETRIC ] that computes a discriminant variety (Lazard and Rouillier, 2007).Computing at least one point per connected component of the semi-algebraic set R \ V ( w H ) using RAGlib gives us points. We evaluate the signatures of H specialized at those points and find that the input system can have , or distinct real solutions when the parameters vary. Remark 23. As we have seen, Algorithm 3 obtains a polynomial which serves similarly as discriminant varieties(Lazard and Rouillier, 2007) or border polynomials (Yang and Xia, 2005) through computing the determinant of para-metric Hermite matrices. Whereas, the two latter strategies rely on algebraic elimination based on Gröbner basesto compute the projection of crit( π, V ) on the y -space. Since it is well-known that the computation of such Gröbnerbasis could be heavy, our algorithm has a chance to be more practical. In Section 8, we provide experimental resultsto support this claim. Remark 24. It is worth noticing that, even though the design of Algorithm 3 employs the grevlex monomial orderingwhere x ≻ · · · ≻ x n , we can replace it by any grevlex ordering with another lexicographical order among the x ’s.For instance, we can use the monomial ordering grevlex ( x n ≻ · · · ≻ x ) . While every theoretical claim still holdsfor this ordering, the practical behavior could be different. We demonstrate this remark in Example 25 below. Example 25. We consider the polynomial sequence ( f , f , f ) ⊂ Q [ y , y , y ][ x , x , x ] f = x x − x ,f = x + 4 x x + 2 x − x x + x x − x + 3 x − x x − x − x − x + 4 x + 4 ,f = y x x + y x + y x + 1 . By computing the reduced Gröbner basis of the ideal generated by f , f , f with respect to the ordering grevlex ( x ≻ x ≻ x ) ≻ grevlex ( y ≻ y ≻ y ) , one note that this system above does not satisfy Assumption ( C ) . Hence, thealgebraic set W ∞ defining the locus over which our parametric Hermite matrix does not well specialize is non-empty.The polynomials w ∞ and w H computed in Algorithm 3 with respect to the monomial ordering grevlex ( x ≻ x ≻ x ) have respectively the degrees and .On the other hand, using the monomial ordering grevlex ( x ≻ x ≻ x ) in Algorithm 3, one obtains a polynomial ˜ w ∞ of degree and the same polynomial w H as above.Therefore, the degree of the input given into the subroutine SamplePoints is reduced by using the second ordering( compared with ). In practice, this choice of ordering accelerates significantly the computation of sample points. .2 Computing semi-algebraic formulas By Corollary 17, the number of real roots of the system f ( η, · ) for a given point η ∈ R t \ W ∞ can be obtained byevaluating the signature of the parametric Hermite matrix H . We recall that the signature of a matrix can be deducedfrom the sign pattern of its leading principal minors. More precisely, we recall the following criterion, introduced bySylvester (1852) and Jacobi (1857) (see Ghys and Ranicki (2016) for a summary on these works). Lemma 26. (Ghys and Ranicki, 2016, Theorem 2.3.6) Let S be a δ × δ symmetric matrix in R δ × δ and, for ≤ i ≤ δ , S i be the i -th leading principal minor of S , i.e., the determinant of the sub-matrix formed by the first i rows and i columns of S . By convention, we denote S = 1 .We assume that S i = 0 for ≤ i ≤ δ . Let k be the number of sign variations between S i and S i +1 . Then, the numbersof positive and negative eigenvalues of S are respectively δ − k and k . Thus, the signature of S is δ − k . This criterion leads us to the following idea. Assume that none of the leading principal minors of H is identically zero.We consider the semi-algebraic subset of R t defined by the non-vanishing of those leading principal minors. Over aconnected component S ′ of this semi-algebraic set, each leading principal minor is not zero and its sign is invariant.As a consequence, by Lemma 26 and Corollary 17, the number of distinct real roots of f ( η, · ) when η varies over S ′ \ W ∞ is invariant.However, this approach does not apply directly if one of the leading principle minors of H is identically zero. Webypass this obstacle by picking randomly an invertible matrix A ∈ GL δ ( Q ) and working with the matrix H A := A T · H · A . The lemma below states that, with a generic matrix A , all of the leading principal minors of H A are notidentically zero. Lemma 27. There exists a Zariski dense subset A of GL δ ( Q ) such that for A ∈ A , all of the leading principal minorsof H A := A T · H · A are not identically zero.Proof. For ≤ r ≤ δ , we denote by M r the set of all r × r minors of H .Let η ∈ Q t \W ∞ ∪W H . We have that H ( η ) is a full rank matrix in Q δ × δ and, for A ∈ GL δ ( R ) , H A ( η ) = A T ·H ( η ) · A .We prove that there exists a Zariski dense subset A of GL δ ( Q ) such that, for A ∈ A , all of the leading principal minorsof H A ( η ) are not zero. Then, as an immediate consequence, all the leading principal minors of H A are not identicallyzero.We consider the matrix A = ( a i,j ) ≤ i,j ≤ δ where a = ( a i,j ) are new variables. Then, the r -th leading principal minor M r ( a ) of A T · H ( η ) · A can be written as M r ( a ) = X m ∈ M r a m · m ( η ) , where the a m ’s are elements of Q [ a ] .As H ( η ) is a full rank symmetric matrix by assumption, there exists a matrix Q ∈ GL δ ( R ) such that Q T · H ( η ) · Q isa diagonal matrix with no zero on its diagonal. Hence, the evaluation of a at the entries of Q gives M r ( a ) a non-zerovalue. As a consequence, M r ( a ) is not identically zero.Let A r be the non-empty Zariski open subset of GL δ ( Q ) defined by M r ( a ) = 0 . Then, the set of the matrices A ∈ A r such that the r × r leading principal minor of A T · H ( η ) · A is not zero.Taking A as the intersection of A r for ≤ r ≤ δ , then, for A ∈ A , none of the leading principal minors of A T ·H ( η ) · A equals zero. Consequently, each leading principal minor of A T · H · A is not identically zero.Our algorithm (Algorithm 4) for solving Problem (1) through parametric Hermite matrices is described below. As itdepends on the random choice of the matrix A , Algorithm 4 is probabilistic. One can easily modify it to be a LasVegas algorithm by detecting the cancellation of the leading principal minors for each choice of A .25 lgorithm 4: RRC-Hermite Input: A polynomial sequence f ⊂ Q [ y ][ x ] such that the ideal h f i is radical and f satisfies Assumption ( A ) Output: The descriptions of a collection of semi-algebraic sets S i solving Problem (1) H , w ∞ ← DRL-Matrix ( f ) Choose randomly a matrix A in Q δ × δ H A ← A T · H · A ( M , . . . , M δ ) ← LeadingPrincipalMinors ( H A ) L ← SamplePoints (cid:0) w ∞ ∧ (cid:0) ∧ δi =1 M i = 0 (cid:1)(cid:1) for η ∈ L do r η ← Signature ( H ( η )) end return { (sign ( M ( η ) , . . . , M δ ( η )) , η, r η ) | η ∈ L } Proposition 28. Assume that f satisfies Assumptions ( A ) and that the ideal h f i is radical. Let A be a matrix in GL δ ( Q ) such that all of the leading principal minors M , . . . , M δ of H A := A T · H · A are not identically zero. Then,Algorithm 4 computes correctly a solution for Problem (1) .Proof. Note that for η ∈ R t \ W ∞ , we have that H A ( η ) = A T · H ( η ) · A . Therefore, the signature of H ( η ) equals tothe signature of H A ( η ) .Let M , . . . , M δ be the leading principal minors of H A and S be the algebraic set defined by ∧ δi =1 M i = 0 . Over eachconnected component S ′ of S , the sign of each M i is invariant and not zero. Therefore, by Lemma 26, the signatureof H A ( η ) , and therefore of H ( η ) , is invariant when η varies over S ′ \ W ∞ . As a consequence, by Corollary 17, thenumber of distinct real roots of f ( η, · ) is also invariant when η varies over S ′ \ W ∞ . We finish the proof of correctnessof Algorithm 4. Example 29. From the parametric Hermite matrix H computed in Example 11, we obtain the sequence of leadingprincipal minors below: M = 4 ,M = 4( − y + y + 2 y ) ,M = 8( − y − y y − y − y y − y y + 2 y ) ,M = 16 y ( − y − y y − y y − y + 3 y y − y y y + 3 y y − y y − y y + y ) . Since M is constant, we compute at least one point per connected component of the semi-algebraic set defined by M = 0 ∧ M = 0 ∧ M = 0 . The computation using RAGlib outputs a set of sample points and finds the following realizable sign conditions of ( M , M , M ) : [ − , , , [ − , − , , [1 , − , − , [ − , − , − , [1 , , − . By evaluating the signature of H at each of those sample points, we deduce the semi-algebraic formulas correspondingto every possible number of real solutions real root → ( M < ∧ M > ∧ M > ∨ ( M < ∧ M < ∧ M > real roots → ( M > ∧ M < ∧ M < ∨ ( M < ∧ M < ∧ M < ∨ ( M > ∧ M > ∧ M < real roots → ( M > ∧ M > ∧ M > . We recall that the semi-algebraic formulas obtained in Example 7 involve the subresultant coefficients s , s and s of degree , and respectively. Whereas, the degrees of the minors M , M and M that we obtain from theparametric Hermite matrix are only , and . In this subsection, we consider an affine regular sequence f = ( f , . . . , f n ) ⊂ Q [ y ][ x ] according to the variables x , i.e., the homogeneous components of largest degree in x of the f i ’s form a homogeneous regular sequence (seeSection 2). Additionally, we require that f satisfies Assumptions ( A ) and ( C ).26et d be the highest value among the total degrees of the f i ’s. Since the homogeneous regular sequences are genericamong the homogeneous polynomial sequences (see, e.g., (Bardet, 2004, Proposition 1.7.4) or Pardue (2010)), thesame property of genericity holds for affine regular sequences (thanks to the definition we use).As in previous sections, G denotes the reduced Gröbner basis of h f i with respect to the ordering grevlex ( x ) ≻ grevlex ( y ) . Let δ be the dimension of the K -vector space K [ x ] / h f i K where K = Q ( y ) . By Bézout’s inequality, δ ≤ d n . We derive from G a basis B = { b , . . . , b δ } of K [ x ] / h f i K consisting of monomials in the variables x . Finally,the parametric Hermite matrix of f with respect to B is denoted by H = ( h i,j ) ≤ i,j ≤ δ .For a polynomial p ∈ Q [ y , x ] , we denote by deg( p ) the total degree of p in ( y , x ) and deg x ( p ) the partial degree of p in the variables x .As Assumption ( C ) holds, by Lemma 13, the entries of the parametric Hermite matrix H associated to f with respectto the basis B are elements of Q [ y ] . To establish a degree bound on the entries of H , we need to introduce the followingassumption. Assumption D . For any g ∈ G , we have that deg( g ) = deg x ( g ) . Proposition 30 below states that Assumption ( D ) is generic. Its direct consequence is a proof for Proposition 14. Proposition 30. Let C [ x , y ] d be the set of polynomials in C [ x , y ] having total degree bounded by d . There exists anon-empty Zariski open subset F D of C [ x , y ] nd such that Assumption ( D ) holds for f ∈ F D ∩ Q [ x , y ] n .Consequently, for f ∈ F D ∩ Q [ x , y ] n , f satisfies Assumption ( C ) .Proof. Let y t +1 be a new indeterminate. For any polynomial p ∈ Q [ x , y ] , we consider the homogenized polynomial p h ∈ Q [ x , y , y t +1 ] of p defined as follows: p h = y deg( p ) t +1 p (cid:18) x y t +1 , . . . , x n y t +1 , y y t +1 , . . . , y t y t +1 (cid:19) . Let C [ x , y , y t +1 ] hd be the set of homogeneous polynomials in C [ x , y , y t +1 ] whose degrees are exactly d . By (Verron,2016, Corollary 1.85), there exists a non-empty Zariski subset F hD of (cid:0) C [ x , y , y t +1 ] hd (cid:1) n such that the variables x is inNoether position with respect to f h for every f h ∈ F hD .For f h ∈ F hD , let G h be the reduced Gröbner basis of f h with respect to the grevlex ordering grevlex ( x ≻ y ≻ y t +1 ) .By (Bardet et al., 2015, Proposition 7), if the variables x is in Noether position with respect to f h , then the leadingmonomials appearing in G h depend only on x .Let f and G be the image of f h and G h by substituting y t +1 = 1 . We show that G is a Gröbner basis of f with respectto the ordering grevlex ( x ≻ y ) .Since G h generates h f h i , G is a generating set of h f i . As the leading monomials of elements in G h do not depend on y t +1 , the substitution y t +1 = 1 does not affect these leading monomials.For a polynomial p ∈ h f i ⊂ Q [ x , y ] , then p writes p = n X i =1 c i · f i , where the c i ’s lie in Q [ x , y ] . We homogenize the polynomials c i · f i on the right hand side to obtain a homogeneouspolynomial P h ∈ h f h i . Note that P h is not necessarily the homogenization p h of p but only the product of p h witha power of y t +1 . Then, there exists a polynomial g h ∈ G h such that the leading monomial of g h divides the leadingmonomial of P h . Since the leading monomial of g h depends only on x , it also divides the leading monomial of p h ,which is the leading monomial of p . So, the leading monomial of the image of g h in G divides the leading monomialof p . We conclude that G is a Gröbner basis of f with respect to the ordering grevlex ( x ≻ y ) and the set of leadingmonomials in G depends only on the variables x .Let F D be the subset of C [ x , y ] nd such that for every f ∈ F D , its homogenization f h is contained in F hD . Since thetwo spaces (cid:0) C [ x , y , y t +1 ] hd (cid:1) n and C [ x , y ] nd are both exactly C ( d + n + tn + t ) × n (by considering each monomial coefficientas a coordinate), F D is also a non-empty Zariski open subset of C [ x , y ] nd .Assume now that the polynomial sequence f belongs to F D . We consider the two monomial orderings over Q [ x , y ] below: 27 The elimination ordering grevlex ( x ) ≻ grevlex ( y ) is abbreviated by O . The leading monomial of p ∈ Q [ x , y ] with respect to O is denoted by lm ( p ) . The reduced Gröbner basis of f with respect to O is G . • The grevlex ordering grevlex ( x ≻ y ) is abbreviated by O . The leading monomial of p ∈ Q [ x , y ] withrespect to O is denoted by lm ( p ) . The reduced Gröbner basis of f with respect to O is denoted by G .As proven above, the set { lm ( g ) | g ∈ G } does not depend on y . With this property, we will show, for any g ∈ G ,there exists a polynomial g ∈ G such that lm ( g ) divides lm ( g ) .By definition, lm ( g ) is greater than any other monomial of g with respect to the ordering O . Since lm ( g ) depends only on the variables x , it is then greater than any monomial of g with respect to the ordering O . Hence, lm ( g ) is also lm ( g ) . Consequently, since G is a Gröbner basis of f with respect to O , there exists a polynomial g ∈ G such that lm ( g ) divides lm ( g ) = lm ( g ) .Next, we prove that for every g ∈ G , lm ( g ) is also lm ( g ) . For this, we rely on the fact that G is reduced. Assumeby contradiction that there exists a polynomial g ∈ G such that lm ( g ) = lm ( g ) . Thus, lm ( g ) must contain both x and y . Let t x be the part in only variables x of lm ( g ) . Note that lm ( g ) is greater than t x with respect to O . Thereexists an element g ∈ G such that lm ( g ) divides lm ( g ) . Since lm ( g ) depends only on the variables x , we havethat lm ( g ) divides t x . Then, by what we proved above, there exists g ′ ∈ G such that lm ( g ) divides lm ( g ) , so lm ( g ) divides t x . This implies that G is not reduced, which contradicts the definition of G .So, lm ( g ) = lm ( g ) for every g ∈ G and, consequently, deg( g ) = deg x ( g ) . We conclude that there exists a non-empty Zariski open subset F D (as above) of C [ x , y ] nd such that Assumption ( D ) holds for every f ∈ F D ∩ Q [ x , y ] n .Additionally, one easily notices that Assumption ( D ) implies Assumption ( C ). As a consequence, f also satisfiesAssumption ( C ) for any f ∈ F D ∩ Q [ x , y ] n .Recall that, when Assumption ( C ) holds, by Lemma 13, the trace of any multiplication map L p is a polynomial in Q [ y ] where p ∈ Q [ y ][ x ] . We now estimate the degree of trace( L p ) . Since the map p trace( L p ) is linear, it issufficient to consider p as a monomial in the variables x . Proposition 31. Assume that Assumption ( D ) holds. Then, for any monomial m in the variables x , the degree in y of trace( L m ) is bounded by deg( m ) . As a consequence, the total degree of the entry h i,j = trace( L b i · b j ) of H is atmost the sum of the total degrees of b i and b j , i.e., deg( h i,j ) ≤ deg( b i ) + deg( b j ) . Proof. Let m be a monomial in Q [ x ] . The multiplication matrix L m is built as follows. For ≤ i ≤ δ , the normalform of b i · m as a polynomial in Q ( y )[ x ] writes NF G ( b i · m ) = δ X j =1 c i,j · b j . Note that this normal form is the remainder of the successive divisions of b i · m by polynomials in G . As Assumption( D ) holds, Assumption ( C ) also holds. Therefore, those divisions do not introduce any denominator. So, every termappearing during these normal form reductions are polynomials in Q [ y ][ x ] .Let p ∈ Q [ y ][ x ] . For any g ∈ G , by Assumption ( D ), the total degree in ( y , x ) of every term of g is at most the degreeof lm x ( g ) . Thus, a division of p by g involves only terms of total degree deg( p ) . Thus, during the polynomial divisionof p to G , only terms of degree at most deg( p ) will appear. Hence the degree of NF G ( p ) is bounded by deg( p ) .Note that trace( L m ) = P δi =1 c i,i . As the degree of c i,i · b i is bounded by deg( b i ) + deg( m ) , the degree of c i,i is atmost deg( m ) . Then, we obtain that deg(trace( L m )) ≤ deg( m ) .Finally, the degree bound of h i,j follows immediately: deg( h i,j ) = deg(trace( L b i · b j )) ≤ deg( b i · b j ) = deg( b i ) + deg( b j ) . Lemma 32. Assume that f satisfies Assumption ( D ) . Then the degree of a minor M consisting of the rows ( r , . . . , r ℓ ) and the columns ( c , . . . , c ℓ ) of H is bounded by ℓ X i =1 (deg( b r i ) + deg( b c i )) . articularly, the degree of det( H ) is bounded by P δi =1 deg( b i ) .Proof. We expand the minors M into terms of the form ( − sign ( σ ) h r ,σ ( c ) . . . h r ℓ ,σ ( c ℓ ) , where σ is a permutationof { c , . . . , c ℓ } and sign ( σ ) is its signature. We then bound the degree of each of those terms as follows usingProposition 31: deg ℓ Y i =1 h r i ,σ ( c i ) ! = ℓ X i =1 deg( h r i ,σ ( c i ) ) ≤ ℓ X i =1 (cid:0) deg( b r i ) + deg( b σ ( c i ) ) (cid:1) = ℓ X i =1 (deg( b r i ) + deg( b c i )) . Hence, taking the sum of all those terms, we obtain the inequality: deg( M i ) ≤ ℓ X i =1 (deg( b r i ) + deg( b c i )) . When M is taken as the determinant of H , then deg(det( H )) ≤ δ X i =1 deg( b i ) . Proposition 31 implies that, when Assumption ( D ) holds, the degree pattern of H depends only on the degree ofthe elements of B = { b , . . . , b δ } . We rearrange B in the increasing order of degree, i.e., deg( b i ) ≤ deg( b j ) for ≤ i < j ≤ δ . So, b = 1 and deg( b ) = 0 . The degree bounds of the entries of H are expressed by the matrix below b ) . . . deg( b δ )deg( b ) 2 deg( b ) . . . deg( b δ ) + deg( b ) ... ... . . . ... deg( b δ ) deg( b δ ) + deg( b ) . . . b δ ) . Moreover, using the regularity of f , we are able to establish explicit degree bounds for the elements of B and then, forthe minors of H . Lemma 33. Assume that f is an affine regular sequence and let B be the basis defined as above. Then the highestdegree among the elements of B is bounded by n ( d − and δ X i =1 deg( b i ) ≤ n ( d − d n . Proof. For p ∈ K [ x ] , let p h ∈ K [ x , . . . , x n +1 ] be the homogenization of p with respect to the variable x n +1 , i.e., p h = x deg x ( p ) n +1 p (cid:18) x x n +1 , . . . , x n x n +1 (cid:19) . The dehomogenization map α is defined as: α : K [ x , . . . , x n +1 ] → K [ x , . . . , x n ] ,p ( x , . . . , x n +1 ) p ( x , . . . , x n , . Also, the homogeneous component of largest degree of p with respect to the variables x is denoted by H p . Throughoutthis proof, we use the following notations:• I = h f i K and G is the reduced Gröbner basis of I w.r.t grevlex ( x ≻ · · · ≻ x n ) .• I h = h p h | p ∈ f i K and G h is the reduced Gröbner basis of I h w.r.t grevlex ( x ≻ · · · ≻ x n +1 ) .29he Hilbert series of the homogeneous ideal I h writes HS I h ( z ) = ∞ X r =0 (dim K K [ x ] r − dim K ( I h ∩ K [ x ] r )) · z r , where K [ x ] r = { p | p ∈ K [ x ] : deg x ( p ) = r } Since f is an affine regular sequence, by definition (see Section 2), H f = ( H f , . . . , H f n ) forms a homoge-neous regular sequence. Equivalently, by (Verron, 2016, Proposition 1.44), the homogeneous polynomial sequence (( f ) h , . . . , ( f n ) h , x n +1 ) is regular. Particularly, (( f ) h , . . . , ( f n ) h ) is a homogeneous regular sequence and, by(Moreno-Socıas, 2003, Theorem 1.5), we obtain HS I h ( z ) = Q ni =1 (cid:0) − z deg( f i ) (cid:1) (1 − z ) n +1 = Q ni =1 (cid:0) . . . + z deg( f i ) − (cid:1) − z . On the other hand, as (( f ) h , . . . , ( f n ) h , x n +1 ) is a homogeneous regular sequence, by (Bardet et al., 2015, Proposi-tion 7), the leading terms of G h w.r.t grevlex ( x ≻ · · · ≻ x n +1 ) do not depend on the variables x n +1 . Thus, thedehomogenization map α does not affect the set of leading terms of G h . Besides, α ( G h ) is a Gröbner basis of I withrespect to grevlex ( x ) (see, e.g., the proof of (Faugère et al., 2013, Lemma 27)). Hence, the leading terms of G h coincides with the leading terms of G .As a consequence, the set of monomials in ( x , . . . , x n +1 ) which are not contained in the initial ideal of I h withrespect to grevlex ( x ≻ · · · ≻ x n +1 ) is exactly { b · x jn +1 | b ∈ B , j ∈ N } . As a consequence, dim K K [ x ] r − dim K ( I h ∩ K [ x ] r ) = r X j =0 |B ∩ K [ x ] j | . Let H ( z ) = P ∞ r =0 |B ∩ K [ x ] r | · z r . We have that (1 − z ) · HS I h ( z ) = (1 − z ) ∞ X r =0 r X j =0 |B ∩ K [ x ] j | · z r = ∞ X r =0 |B ∩ K [ x ] r | · z r = H ( z ) . Then, H ( z ) = n Y i =1 (cid:16) . . . + z deg( f i ) − (cid:17) . As a direct consequence, max ≤ i ≤ δ deg( b i ) is bounded by P ni =1 deg( f i ) − n ≤ n ( d − .Let G and G be two polynomials in Z [ z ] . We write G ≤ G if and only if for any r ≥ , the coefficient of z r in G is greater than or equal to the one in G .Since deg( f i ) ≤ d for every ≤ i ≤ n , then H ( z ) = n Y i =1 (cid:16) . . . + z deg( f i ) − (cid:17) ≤ n Y i =1 (cid:0) . . . + z d − (cid:1) . As a consequence, H ′ ( z ) = ∞ X r =1 ( r |B ∩ K [ x ] r | ) · z r − ≤ n Y i =1 (cid:0) . . . + z d − (cid:1)! ′ . Expanding G ′ ( z ) , we obtain H ′ ( z ) ≤ n (cid:0) . . . + z d − (cid:1) n − (cid:0) . . . + z d − − dz d − (cid:1) − z = n (cid:0) . . . + z d − (cid:1) n − d − X i =0 z i − z d − − z = n (cid:0) . . . + z d − (cid:1) n − d − X i =0 z i (cid:0) . . . + z d − i − (cid:1) . 30y substituting z = 1 in the above inequality, we obtain H ′ (1) ≤ nd n − d − X i =0 ( d − i − 1) = n ( d − d n . Thus, we have that δ X i =1 deg( b i ) = ∞ X r =0 r |B ∩ K [ x ] r | = H ′ (1) ≤ n ( d − d n . Corollary 34 below follows immediately from Lemmas 32 and 33. Corollary 34. Assume that f is a regular sequence that satisfies Assumption ( D ) . Then the degree of any minor of H is bounded by n ( d − d n . Example 35. We consider again the system f = ( x + x − y , x x + y x + y x ) in Example 11. Note that f forms a regular sequence.The Gröbner basis G of f with respect to the ordering grevlex ( x ) ≻ grevlex ( y ) is G = { x + y x + ( y − y ) x + y y x − y y , x + x − y , x x + x y + x y } . So, f satisfies Assumption ( D ) . The matrix with respect to the basis B = { , x , x , x } has the following degreepattern: This degree pattern agrees with the result of Proposition 31. The determinant of this matrix is of degree , which isindeed smaller than n ( d − d n = 8 Whereas, using the basis B = { , x , x , x } leads to another parametric Hermite matrix of different degrees. For ≤ i, j ≤ , the degree of its ( i, j ) -entry, which is equals to trace( L x i + j − ) , is bounded by deg( x i − ) + deg( x j − ) = i + j − using Proposition 31. Applying Lemma 32, the determinant is bounded by P i =0 deg( x i ) = 12 .By computing the parametric Hermite matrix of f with respect to B , we obtain the degree pattern on its entries and a determinant of degree . Again, both of our theoretical bounds hold for this matrix. Remark 36. Note that Assumption ( D ) requires a condition on the degrees of polynomials in the Gröbner basis G of h f i . We remark that it is possible to establish similar bounds for the degrees of entries of our parametric Hermitematrix and its minors when the system f satisfies a weaker property than Assumption ( D ) (we still keep the regularityassumption).Indeed, we only need to assume that, for any g ∈ G , the homogeneous component of the highest degree in x of g doesnot depend on the parameters y . Let d y be an upper bound of the partial degrees in y of elements of G . Under thechange of variables x i x d y i , f is mapped to a new polynomial sequence that satisfies Assumption ( D ) . Therefore,we easily deduce the two following bounds, which are similar to the ones of Proposition 31 and Corollary 34. • deg( h i,j ) ≤ d y (deg( b i ) + deg( b j )) ; • The degree of any minor of H is bounded by d y n ( d − d n .Even though these bounds are not sharp anymore, they still allow us to compute the parametric Hermite matrices usingevaluation & interpolation scheme and control the complexity of this computation in the instances where Assumption ( D ) does not hold. .2 Complexity analysis of our algorithms In this subsection, we analyze the complexity of our algorithms on generic systems.Let f = ( f , . . . , f n ) ⊂ Q [ x , y ] be a regular sequence, where y = ( y , . . . , y t ) and x = ( x , . . . , x n ) , satis-fying Assumptions ( A ) and ( D ). We denote by G be the reduced Gröbner basis of f with respect to the ordering grevlex ( x ) ≻ grevlex ( y ) . The basis B is taken as all the monomials in x that are irreducible by G . Then, H is theparametric Hermite matrix associated of f with respect to B .We start by estimating the arithmetic complexity for computing the parametric Hermite matrix H and its minors. Wedenote λ := n ( d − and D := n ( d − d n . Proposition 37. Assume that f = ( f , . . . , f n ) ⊂ Q [ y ][ x ] is a regular sequence that satisfies Assumptions ( A ) and ( D ) . Then, the following holds.i) The parametric Hermite matrix H can be computed using O e(cid:18)(cid:18) t + 2 λt (cid:19) (cid:18) n (cid:18) d + n + tn + t (cid:19) + δ ω +1 + δ log (cid:18) t + 2 λt (cid:19)(cid:19)(cid:19) arithmetic operations in Q .ii) Each minor (including the determinant) of H can be computed using O e(cid:18)(cid:18) t + D t (cid:19) (cid:18) δ (cid:18) t + 2 λt (cid:19) + δ ω + log (cid:18) t + D t (cid:19)(cid:19)(cid:19) arithmetic operations in Q .Proof. For the computation of the matrix, we rely on Proposition 19 which estimates the complexity of the evaluation& interpolation scheme described in Subsection 5.4.By Lemma 33 and Proposition 31, the highest degree among the entries of H is bounded by λ = 2 n ( d − . Therefore,we replace Λ in Proposition 19 by λ in the complexity statement of Proposition 19 to obtain O e(cid:18)(cid:18) t + 2 λt (cid:19) (cid:18) n (cid:18) d + n + tn + t (cid:19) + δ ω +1 + δ log (cid:18) t + 2 λt (cid:19)(cid:19)(cid:19) . Similarly, the minors of H can be computed using the technique of evaluation & interpolation.By Corollary 34, the degree of every minor of H is bounded by D . We specialize H at (cid:0) t + D t (cid:1) points in Q t and computethe corresponding minor of each specialized Hermite matrix. This step takes O (cid:18)(cid:18) t + D t (cid:19) (cid:18) δ (cid:18) t + 2 λt (cid:19) + δ ω (cid:19)(cid:19) arithmetic operations in Q . Finally, using the multivariate interpolation algorithm of Canny et al. (1989), it requires O e(cid:18)(cid:18) t + D t (cid:19) log (cid:18) t + D t (cid:19)(cid:19) arithmetic operations in Q to interpolate the final minor. Therefore, the whole complexity for computing each minorof H lies within O e(cid:18)(cid:18) t + D t (cid:19) (cid:18) δ (cid:18) t + 2 λt (cid:19) + δ ω + log (cid:18) t + D t (cid:19)(cid:19)(cid:19) . Finally, we state our main result, which is Theorem II below. It estimates the arithmetic complexity of Algorithms 3and 4. Theorem II. Let f ⊂ Q [ x , y ] be a regular sequence such that the ideal h f i is radical and f satisfies Assumptions ( A ) and ( D ) . Recall that D denotes n ( d − d n . Then, we have the following statements:i) The arithmetic complexity of Algorithm 3 lies in O e(cid:18)(cid:18) t + D t (cid:19) t n t +1 d nt + n +2 t +1 (cid:19) . i) Algorithm 4, which is probabilistic, computes a set of semi-algebraic descriptions solving Problem (1) within O e(cid:18)(cid:18) t + D t (cid:19) t n t +1 d nt +2( n + t )+1 (cid:19) arithmetic operations in Q in case of success.iii) The semi-algebraic descriptions output by Algorithm 4 consist of polynomials in Q [ y ] of degree bounded by D .Proof. As Assumption ( D ) holds, we have that w ∞ = 1 and w H is the square-free part of det( H ) .Therefore, after computing the parametric Hermite matrix H and its determinant, whose complexity is given by Propo-sition 37, Algorithm 3 essentially consists of computing sample points of the connected components of the algebraicset R t \ V (det( H )) .By Corollary 34, the degree of det( H ) is bounded by D . Applying Corollary 4, we obtain the following arithmeticcomplexity for this computation of sample points O e(cid:18)(cid:18) t + D t (cid:19) t t D t +1 (cid:19) ≃ O e(cid:18)(cid:18) t + D t (cid:19) t n t +1 d nt + n +2 t +1 (cid:19) . Also by Corollary 4, the finite subset of Q t output by SamplePoints has cardinal bounded by t D t . Thus, evaluatingthe specializations of H at those points and their signatures costs in total O (cid:16) t D t (cid:16) δ (cid:0) λ + tt (cid:1) + δ ω +1 / (cid:17)(cid:17) arithmeticoperations in Q using (Basu et al., 2006, Algorithm 8.43).Therefore, the complexity of SamplePoints dominates the whole complexity of the algorithm. We conclude thatAlgorithm 3 runs within O e(cid:18)(cid:18) t + D t (cid:19) t n t +1 d nt + n +2 t +1 (cid:19) arithmetic operations in Q .For Algorithm 4, we start by choosing randomly a matrix A and compute the matrix H A = A T · H · A . Then,we compute the leading principal minors M , . . . , M δ of H A . Using Proposition 37, this step admits the arithmeticcomplexity bound O e(cid:18) δ (cid:18) t + D t (cid:19) (cid:18) δ (cid:18) t + 2 λt (cid:19) + δ ω + log (cid:18) t + D t (cid:19)(cid:19)(cid:19) . Next, Algorithm 4 computes sample points for the connected components of the semi-algebraic set defined by ∧ δi =1 M i = 0 . Since the degree of each M i is bounded by D , Corollary 4 gives the arithmetic complexity O e(cid:18)(cid:18) t + D t (cid:19) t d nt + n t D t +1 (cid:19) ≃ O e(cid:18)(cid:18) t + D t (cid:19) t n t +1 d nt +2( n + t )+1 (cid:19) . It returns a finite subset of Q t whose cardinal is bounded by (2 δ D ) t . The evaluation of the leading principal minors’sign patterns at those points has the arithmetic complexity lying in O (cid:0) t δ t +1 D t (cid:1) ≃ O (cid:0) t n t d nt + n +2 t (cid:1) .Again, the complexity of SamplePoints dominates the whole complexity of Algorithm 4. The proof of Theorem II isthen finished. Recall that Algorithm 4 leads us to compute sample points per connected components of the non-vanishing set ofthe leading principal minors ( M , . . . , M δ ) . Comparing to Algorithm 3 in which we only compute sample points for R t \ V ( M δ ) , the complexity of Algorithm 4 contains an extra factor of d nt due to the higher number of polynomialsgiven as input to the subroutine SamplePoints . Even though the complexity bounds of these two algorithms bothlie in d O ( nt ) , the extra factor d nt mentioned above sometimes becomes the bottleneck of Algorithm 4 for tacklingpractical problems. Therefore, we introduce the following optimization in our implementation of Algorithm 4.We start by following exactly the steps (1-4) of Algorithm 4 to obtain the leading principal minors ( M , . . . , M δ ) andthe polynomial w ∞ . Then, by calling the subroutine SamplePoints on the input M δ = 0 ∧ w ∞ = 0 , we compute a set33f sample points (and their corresponding numbers of real roots) { ( η , r ) , . . . , ( η ℓ , r ℓ ) } that solves the weak-versionof Problem (1). We obtain from this output all the possible numbers of real roots that the input system can admit.For each value ≤ r ≤ δ , we define Φ r = { σ = ( σ , . . . , σ δ ) ∈ {− , } δ | the sign variation of σ is ( δ − r ) / } . If r δ (mod 2) , Φ r = ∅ .For σ ∈ Φ r and η ∈ R t \ V ( w ∞ ) such that sign ( M i ( η )) = σ i for every ≤ i ≤ δ , the signature of H ( η ) is r . As aconsequence, for any η in the semi-algebraic set defined by ( w ∞ = 0) ∧ ( ∨ σ ∈ Φ r ( ∧ δi =1 sign ( M i ) = σ i )) , the system f ( η, . ) has exactly r distinct real solutions.Therefore, ( S r i ) ≤ i ≤ ℓ is a collection of semi-algebraic sets solving Problem (1). Then, we can simply return { (Φ r i , η i , r i ) | ≤ i ≤ ℓ } as the output of Algorithm 4 without any further computation. Note that, by doingso, we may return sign conditions which are not realizable.We discuss now about the complexity aspect of the steps described above. For r ≡ δ (mod 2) , the cardinal of Φ r is (cid:0) δ ( δ − r − / (cid:1) . In theory, the total cardinal of all the Φ r i ’s ( ≤ i ≤ ℓ ) can go up to δ − , which is doubly exponentialin the number of variables n . However, in the instances that are actually tractable by the current state of the art, δ is still smaller than δ t . And when it is the case, following this approach has better performance than computing thesample points of the semi-algebraic set defined by ∧ δi =1 M i = 0 . Otherwise, when δ exceeds δ t , we switch back tothe computation of sample points.This implementation of Algorithm 4 does not change the complexity bound given in Theorem II. This subsection provides numerical results of several algorithms related to the real root classification. We report onthe performance of each algorithm for different test instances.The computation is carried out on a computer of Intel(R) Xeon(R) CPU E7-4820 2GHz and 1.5 TB of RAM. Thetimings are given in seconds (s.), minutes (m.) and hours (h.). The symbol ∞ means that the computation cannotfinish within hours.Throughout this subsection, the column HERMITE reports on the computational data of our algorithms based on para-metric Hermite matrices described in Section 6. It uses the notations below:- MAT : the timing for computing a parametric Hermite matrix H .- DET : the runtime for computing the determinant of H .- MIN : the timing for computing the leading principal minors of H .- SP : the runtime for computing at least one points per each connected component of the semi-algebraic set R t \ V (det( H )) .- DEG : the highest degree among the leading principal minors of H . Generic systems In this paragraph, we report on the results obtained with generic inputs, i.e., randomly chosendense polynomials ( f , . . . , f n ) ⊂ Q [ y , . . . , y t ][ x , . . . , x n ] . The total degrees of input polynomials are given as alist d = [deg( f ) , . . . , deg( f n )] .We first compare the algorithms using Hermite matrices (Section 6) with the Sturm-based algorithm (Section 4) forsolving Problem (1). The column STURM of Fig. (1) shows the experimental results of the Sturm-based algorithm. Itcontains the following sub-columns:- ELIM : the timing for computing the eliminating polynomial.- SRES : the timing for computing the subresultant coefficients in the Sturm-based algorithm.- SP - S : the timing for computing sample points per connected components of the non-vanishing set of the lastsubresultant coefficient.- DEG - S : the highest degree among the subresultant coefficients.34e observe that the sum of MAT - H and MIN - H is smaller than the sum of ELIM and SRES . Hence, obtaining theinput for the sample point computation in HERMITE strategy is easier than in STURM strategy. We also remark thatthe degree DEG - H is much smaller than DEG - S , that explains why the computation of sample points using Hermitematrices is faster than using the subresultant coefficients.We conclude that the parametric Hermite matrix approach outperforms the Sturm-based one both on the timings andthe degree of polynomials in the output formulas. t d HERMITE STURMMAT MIN SP total DEG ELIM SRES SP - S total DEG - S , .07 s .01 s .3 s .4 s 8 .01 s .1 s 2 s 2.2 s 12 , .1 s .12 s 4.8 s 5 s 18 .05 s .5 s 15 s 16 s 30 , , .3 s .3 s 33 s 34 s 24 .08 s 2 s 8 m 8 m 56 , .3 s .8 s 3 m 3 m 36 .1 s 3 s 20 m 20 m 72 , .1 s .02 s 26 s 27 s 8 .07 s .1 s 40 s 40 s 12 , .2 s .2 s 3 h 3 h 18 .1 s 1 s ∞ ∞ , , .5 s 7 s 32 h 32 h 24 .15 s 10 m ∞ ∞ , .6 s 12 s 90 h 90 h 32 .2 s 12 m ∞ ∞ , ∞ ∞ 36 .2 s 15 m ∞ ∞ Figure 1: Generic random dense systemsIn Fig. (2), we compare our algorithms using parametric Hermite matrices with two Maple packages forsolving parametric polynomial systems: R OOT F INDING [P ARAMETRIC ] (Gerhard et al., 2010) and R EGULAR -C HAINS [P ARAMETRIC S YSTEM T OOLS ] (Yang et al., 2001). The new notations used in Fig. (2) are explained below.• The column RF stands for the R OOT F INDING [P ARAMETRIC ] package. To solve a parametric polynomialsystems, it consists of computing a discriminant variety D and then computing an open CAD of R t \ D . Thispackage does not return explicit semi-algebraic formulas but an encoding based on the real roots of somepolynomials. This column contains:- DV : the runtime of the command D ISCRIMINANT V ARIETY that computes a set of polynomials defininga discriminant variety D associated to the input system.- CAD : the runtime of the command C ELL D ECOMPOSITION that outputs semi-algebraic formulas bycomputing an open CAD for the semi-algebraic set R t \ D . • The column RC stands for the R EGULAR C HAINS [P ARAMETRIC S YSTEM T OOLS ] package of Maple. Thealgorithms implemented in this package is given in Yang et al. (2001). It also contains two sub-columns:- BP : the runtime of the command B ORDER P OLYNOMIAL that returns a set of polynomials.- RRC : the runtime of the command R EAL R OOT C LASSIFICATION . We call this command with the option output=‘samples’ to compute at least one point per connected component of the complementary ofthe real algebraic set defined by border polynomials.Note that, in a strategy for solving the weak-version of Problem (1), D ISCRIMINANT V ARIETY and B ORDER P OLYNO - MIAL can be completely replaced by parametric Hermite matrices.On generic systems, the determinant of our parametric Hermite matrix coincides with the output of D ISCRIMINANT -V ARIETY , which we denote by w . Whereas, because of the elimination B ORDER P OLYNOMIAL returns several poly-nomials, one of them is w .In Fig. (2), the timings for computing a parametric Hermite matrix is negligible. Comparing the columns DET , DV and BP , we remark that the time taken to obtain w through the determinant of parametric Hermite matrices is muchsmaller than using D ISCRIMINANT V ARIETY or B ORDER P OLYNOMIAL .For computing the polynomial w , using parametric Hermite matrices allows us to reach the instances that are outof reach of D ISCRIMINANT V ARIETY , for example, the instances { t = 3 , d = [2 , , } , { t = 3 d = [4 , } , { t = 3 , d = [3 , } and { t = 4 , d = [2 , } in Fig. (2) below. Moreover, we succeed to compute the semi-algebraicformulas for { t = 3 , d = [2 , , } , { t = 3 d = [4 , } and { t = 4 , d = [2 , } . Using the implementation inSubsection 8.1, we obtain the semi-algebraic formulas of degrees bounded by deg( w ) .Therefore, for these generic systems, our algorithm based on parametric Hermite matrices outperforms D ISCRIMI - NANT V ARIETY and B ORDER P OLYNOMIAL for obtaining a polynomial that defines the boundary of semi-algebraic35ets over which the number of real solutions are invariant. Moreover, using the minors of parametric Hermite ma-trices, we can compute semi-algebraic formulas of problems that are out of reach of C ELL D ECOMPOSITION andR EAL R OOT C LASSIFICATION . t d HERMITE RF RCMAT DET SP total DEG DV CAD total BP RRC total , .07 s .01 s .3 s .4 s 8 .1 s .3 s .4 s .1 s 1 s 1.1 s , .1 s .2 s 4.8 s 5 s 18 1 m 5 s 1 m .3 s 12 s 12 s , , .3 s .3 s 33 s 34 s 24 17m 32 s 17m 23 s 2 m 2 m , .3 s .8 s 3 m 3 m 36 2 h 4 m 2 h 8 s 4 m 4 m , .1 s .02 s 26 s 27 s 8 1 s 35 s 36 s .2 s 12m 12m , .2 s .2 s 3 h 3 h 18 2 h 84 h 86 h 3 s 37 h 37 h , , .5 s 7 s 32 h 32 h 24 ∞ ∞ ∞ 20 m ∞ ∞ , .6 s 12 s 90 h 90 h 32 ∞ ∞ ∞ 12 m ∞ ∞ , .7 s 27 s ∞ ∞ ∞ ∞ ∞ 15 m ∞ ∞ , .2 s .1 s 8 m 8 m 8 4 s ∞ ∞ ∞ ∞ Figure 2: Generic random dense systems36n what follows, we consider the systems coming from some applications as test instances. These examples allow usto observe the behavior of our algorithms on non-generic systems. Kuramoto model This application is introduced in Kuramoto (1975), which is a dynamical system used to modelsynchronization among some given coupled oscillators. Here we consider only the model constituted by oscillators.The maximum number of real solutions of steady-state equations of this model was an open problem before it is solvedin Harris et al. (2020) using numerical homotopy continuation methods. However, to the best of our knowledge, thereis no exact algorithm that is able to solve this problem. We present in what follows the first solution using symboliccomputation. Moreover, our algorithm can return the semi-algebraic formulas defining the regions over which thenumber of real solutions is invariant.As explained in Harris et al. (2020), we consider the system f of the following equations (cid:26) y i − P j =1 ( s i c j − s j c i ) = 0 s i + c i = 1 for ≤ i ≤ , where ( s , s , s ) and ( c , c , c ) are variables and ( y , y , y ) are parameters. We are asked to compute the maximumnumber of real solutions of f ( η, . ) when η varies over R . This leads us to solve the weak version of Problem (1) forthis parametric system.We first construct the parametric Hermite matrix H associated to this system. This matrix is of size × . Thepolynomial w ∞ has the factors y + y , y + y , y + y and y + y + y . The polynomial w H has degree (c.f.Harris et al. (2020)). We denote by w the polynomial w ∞ · w H .Note that the polynomial system has real roots only if | y i | ≤ (c.f. Harris et al. (2020)). So we only need to considerthe compact connected components of R \ V ( w ) . Since the polynomial w is invariant under any permutation actingon ( y , y , y ) , we exploit this symmetry to accelerate the computation of sample points.Following the critical point method, we compute the critical points of the map ( y , y , y ) y + y + y restrictedto R \ V ( w ) ; this map is also symmetric. We apply the change of variables ( y , y , y ) ( e , e , e ) , where e = y + y + y , e = y y + y y + y y and e = y y y are elementary symmetric polynomials of ( y , y , y ) . This change of variables reduces the number of distinct solutions of zero-dimensional systems involvedin the computation and, therefore, reduces the computation time.From the sample points obtained by this computation, we derive the possible number of real solutions and concludethat the system f has at most distinct real solutions when ( y , y , y ) varies over R .Fig. (3) reports on the timings for computing the parametric Hermite matrix ( MAT ), for computing its determinant( DET ) and for computing the sample points ( SP ). We stop both of the commands D ISCRIMINANT V ARIETY andB ORDER P OLYNOMIAL after hours without obtaining the polynomial w . HERMITE DV BPMAT DET SP total m h h 86 h ∞ ∞ Figure 3: Kuramoto model for oscillators Static output feedback The second non-generic example comes from the problem of static output feedback(Henrion and Sebek, 2008). Given the matrices A ∈ R ℓ × ℓ , B ∈ R ℓ × , C ∈ R × ℓ and a parameter vector P = (cid:20) y y (cid:21) ∈ R , the characteristic polynomial of A + BP C writes f ( s, y ) = det( sI l − A − BKC ) = f ( s ) + y f ( s ) + y f ( s ) , where s is a complex variable.We want to find a matrix P such that all the roots of f ( s, y ) must lie in the open left half-plane. By substituting s by x + ix , we obtain the following system of real variables ( x , x ) and parameters ( y , y ) : ( ℜ ( f ( x + ix , y )) = 0 ℑ ( f ( x + ix , y )) = 0 x < ℓ .We are now interested in solving the weak-version of Problem (1) on the system ℜ ( f ) = ℑ ( f ) = 0 . We observe thatthis system satisfies Assumptions ( A ) and ( C ). Let H be the parametric Hermite matrix H of this system with respectto the usual basis we consider in this paper. This matrix H behaves very differently from generic systems.Computing the determinant of H (which is an element of Q [ y ] ) and taking its square-free part allows us to obtainthe same output w as D ISCRIMINANT V ARIETY . However, this direct approach appears to be very inefficient as thedeterminant appears as a large power of the output polynomial.For example, for a value ℓ , we observe that the system consists of two polynomials of degree ℓ . The determinant of H appears as w ℓ , where w has degree ℓ − . The bound we establish on the degree of this determinant is ℓ − ℓ ,which is much larger than what happens in this case. Therefore, we need to introduce the optimization below to adaptour implementation of Algorithm 3 to this problem.We observe that, on these examples, the polynomial w can be extracted from a smaller minor instead of computingthe determinant H . To identify such a minor, we reduce H to a matrix whose entries are univariate polynomials withcoefficients lying in a finite field Z /p Z as follow.Let u be a new variable. We substitute each y i by random linear forms in Q [ u ] in H and then compute H mod p .Then, the matrix H is turned into a matrix H u whose entries are elements of Z /p Z [ u ] . The computation of the leadingprincipal minors of H u is much easier than the one of H since it involves only univariate polynomials and does notsuffer from the growth of bit-sizes as for the rational numbers.Next, we compute the sequence of the leading principal minors of H u in decreasing order, starting from the determi-nant. Once we obtain a minor, of some size r , that is not divisible by w u , we stop and take the index r + 1 . Then,we compute the square-free part of the ( r + 1) × ( r + 1) leading principal minor of H , which can be done throughevaluation-interpolation method. This yields a Monte Carlo implementation that depends on the choice of the randomlinear forms in Q [ u ] and the finite field to compute the polynomial w .In Fig. (4), we report on some computational data for the static output feedback problem. Here we choose theprime p to be so that the elements of the finite field Z /p Z can be represented by a machine word of bits.We consider different values of ℓ and the matrices A, B, C are chosen randomly. On these examples, our algorithmreturns the same output as the one of D ISCIMINANT V ARIETY . Whereas, B ORDER P OLYNOMIAL ( BP ) returns a list ofpolynomials which contains our output and other polynomials of higher degree.The timings of our algorithm are given by the two following columns:• The column MAT shows the timings for computing parametric Hermite matrices H .• The column COMP - W shows the timings for computing the polynomials w from H using the strategy de-scribed as above.We observe that our algorithm ( MAT + COMP - W ) wins some constant factor comparing to D ISCRIMINANT V ARIETY ( DV ). On the other hand, B ORDER P OLYNOMIAL ( BP ) performs less efficiently than the other two algorithms in theseexamples.Since the degrees of the polynomials w here (given as DEG - W ) are small comparing with the bounds in the genericcase. 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