Critical Point Computations on Smooth Varieties: Degree and Complexity bounds
aa r X i v : . [ c s . S C ] M a y Critical Point Computations on Smooth Varieties:Degree and Complexity bounds
Mohab Safey El DinSorbonne Universit´es, UPMC Univ Paris 06, 7606, LIP6, F-75005, Paris, FranceCNRS, UMR 7606, LIP6, F-75005, Paris, FranceInria, Paris Center, PolSys ProjectPierre-Jean SpaenlehauerInria, CNRS, Universit´e de LorraineNancy, France
Abstract
Let V ⊂ C n be an equidimensional algebraic set and g be an n -variate polynomial withrational coefficients. Computing the critical points of the map that evaluates g at the pointsof V is a cornerstone of several algorithms in real algebraic geometry and optimization. Underthe assumption that the critical locus is finite and that the projective closure of V is smooth,we provide sharp upper bounds on the degree of the critical locus which depend only on deg( g )and the degrees of the generic polar varieties associated to V . Hence, in some special caseswhere the degrees of the generic polar varieties do not reach the worst-case bounds, this impliesthat the number of critical points of the evaluation map of g is less than the currently knowndegree bounds. We show that, given a lifting fiber of V , a slight variant of an algorithm due toBank, Giusti, Heintz, Lecerf, Matera and Solern´o computes these critical points in time whichis quadratic in this bound up to logarithmic factors, linear in the complexity of evaluating theinput system and polynomial in the number of variables and the maximum degree of the inputpolynomials. Problem statement.
Let f = ( f , . . . , f p ) ⊂ Q [ X , . . . , X n ] be a polynomial system defining asmooth and equidimensional algebraic set V ⊂ C n of dimension d and g ∈ Q [ X , . . . , X n ] be apolynomial of degree D . We focus on the complexity of computing the critical points of the mapevaluating g at the points of V . These critical points are defined by f = · · · = f p = 0 and by thesimultaneous vanishing of the ( n − d + 1)-minors of the jacobian matrix jac( f , g ) ∂f ∂X · · · ∂f ∂X n ... ... ∂f p ∂X · · · ∂f p ∂X n ∂g∂X · · · ∂g∂X n . q, v , . . . , v n ) , λ ) where q and the v i ’s lie in Q [ T ] ( T is a new variable) and λ is a linear form in X , . . . , X n with q square-free, deg( v i ) < deg( q ), λ ( v , . . . , v n ) = T ∂q∂T mod q and the set defined by q ( τ ) = 0 , X i = v i ∂q/∂T ( τ ) 1 ≤ i ≤ n coincides with the critical locus under consideration. Observe that the degree of q coincides withthe number of critical points and the number of rational numbers required in such a rationalparametrization is O ( n deg( q )).Assuming again the critical locus to be finite, several bounds on its cardinality have been established(see [26] and references therein). These bounds depend on n , p and the degrees of f , . . . , f p and g . However, it has been remarked that when V is not a complete intersection, or when it hassome special properties, the cardinality of the critical locus may be far less than these bounds andsometimes depends only on D and on some quantities attached to geometric objects. These latterobjects are polar varieties (see [2, 4]); they may be understood as the critical loci of the restrictionto V of projections on generic linear subspaces ; we define them further precisely.Assuming the smoothness of the projective closure of V and the finiteness of the critical locus understudy, this paper adresses the following topical questions. • Provide a bound on the number of complex critical points depending on D and on the degreesof the generic polar varieties associated to V . • Find an algorithm computing a rational parametrization of this critical locus within an arith-metic complexity which is essentially quadratic in the obtained bound and polynomial in p , n , the complexity of evaluating f , g and max ≤ i ≤ p (deg( f i )). Motivations and prior works.
Since local extrema of the evaluation map of g are reached atcritical points, computing critical points is a basic and useful task for polynomial optimization(see e.g. [18, 19, 30]). Because of their topological properties related to Morse theory, computingcritical points is also a subroutine for many modern algorithms in real algebraic geometry yieldingasymptotically optimal complexities (see e.g. [20, 21, 29] and [6] for a textbook reference on thisfamily of algorithms). Polar varieties have been introduced in [2] for computing sample points ineach connected component of a real algebraic set and this technique has been developed in [3, 31];they are also used for computing roadmaps for deciding connectivity queries [7, 8, 32, 33], forcomputing the real dimension of a real algebraic set (see [5] and references therein) or for variantquantifier elimination [23].Some bounds on the cardinality of the critical locus under consideration are given in [26] when f , . . . , f p is a regular sequence. These bounds depend on the degrees of the f i ’s, D , n and p .Since polynomial systems appearing in applications arise most of the time with a special structure,a natural question to ask for is to identify situations where the cardinality of the considered criticallocus is less than what the worst case bounds predicted in [26].Such situations have been exhibited in [9] where critical points are used in computational statisticsvia the notion of ML degree. When deg( g ) = 2 the cardinality of the critical locus is boundedby the generic ED degree of V which depends only on the degrees of the generic polar varietiesassociated to V [10]. These bounds do not require any smoothness assumption. The results onpolar varieties in [22, 28] play a central role in this setting.2n the algorithmic side, many recent works have focused on the complexity of computing criticalloci. The results in [14, 35] provide complexity bounds for computing critical points using Gr¨obnerbases under genericity assumptions on the input polynomials f . The obtained complexity boundsare not quadratic in the generic number of critical points and the genericity assumptions are notwell-suited to the situations we are willing to consider. The results in [1] provide complexity boundsfor a probabilistic algorithm computing degeneracy loci in time quadratic in an intrinsic quantitycalled the system degree . This work is strongly related to the algorithmic framework of the solverproposed in [17] and to computational aspects of polar varieties which have been deeply investigatedin the last decades, see [2, 3, 4] and references therein. We will use a slight variant of [1] for ouralgorithmic contribution. Main results.
Under some smoothness assumptions which are precised below, we prove a boundon the number of complex critical points of the map x ∈ V → g ( x ) depending on the degree of g and integers δ ( V ) , . . . , δ d +1 ( V ). The number δ i +1 ( V ) is the degree of the polar variety of V associated to a generic linear projection on C i +1 . In the sequel, given a = ( a , . . . , a i ) ⊂ C ni , wedenote by W (( g, a ) , V ) the critical locus of the map x ∈ V → ( g ( x ) , a · x, . . . , a i · x ). We alsodenote by C [ X , . . . , X n ] ≤ D the set of polynomials in C [ X , . . . , X n ] of total degree ≤ D . Theorem 1.
Let V ⊂ C n be a d -equidimensional algebraic set whose projective closure is smoothand D ≥ and i ∈ { , . . . , d } . There exists a Zariski dense subset Ω i ⊂ C [ X , . . . , X n ] ≤ D × C i × n such that for any ( g, a ) ∈ Ω , the degree of W (( g, a ) , V ) ⊂ C n is bounded by deg( W (( g, a ) , V )) ≤ ( δ i +1 ( V ) if D = 1 P dj = i δ j +1 ( V )( D − j − i otherwise . One of the central ingredient of the proof is an algebraic version of Thom’s weak transversalityTheorem. We use a formalism and notation similar to [33, Sec 4.2] which provides a proof ofthis result using charts and atlases. We also show that this degree bound holds under milderassumptions on g : it is sufficient to assume that the evaluation map of g has finitely many criticalpoints on V . Let us see how such a bound behaves on an example. Example 2.
Let V ⊂ C be the set of points ( x , . . . , x ) where the matrix x x x x x x x x has rank . This variety has dimension , degree and ( δ , . . . , δ ) = (1 , , , , . Consider g = P i =1 ix i . Representing an open subset of V as the zero locus of a reduced regular sequenceof quadratic polynomials f , . . . , f , bounds depending on the degrees of f , . . . , f , g (see e.g. [26,Thm. 2.2]) would give the upper bound for the number of complex critical points. Theorem1 and its variant for non-generic objective functions (see Prop. 12 below) yield the bound
241 = δ + 2 δ + 4 δ + 8 δ + 16 δ . Computations show that the evaluation map of g restricted to V hasactually exactly complex critical points on V . A convenient representation of an equidimensional variety V of dimension d is a lifting fiber for V (see [17]). Roughly speaking, this lifting fiber consists in a rational parametrization of the (finite)set of points in a section of V by a ( n − d )-dimensional affine plane, together with a lifting system V has been precomputed and that deg( g ) ≥
2, we use the algorithm proposedin [1] to compute the critical points. Since this algorithm handles the more general case of quasi-affine varieties, so does the proposed variant. However, our main complexity results hold only underthe assumption that the projective closure of V is smooth and that the evaluation map of g hasfinitely many critical points on V . Our second main result is a proof that the arithmetic complexityis quadratic up to logarithmic factors in the degree bound from Theorem 1, polynomial in n , themaximum of the degrees of the lifting system, deg( g ), and the complexity of evaluating the liftingsystem and g . Organization of the paper.
Section 2 describes notation and preliminary results used throughoutthis paper. Section 3 is devoted to the proof of Theorem 1. It relies on a transversality result whichis proved in Section 4. Section 5 deals with nongeneric objective functions. Finally, Section 6discusses algorithmic aspects and complexity bounds.
Acknowledgments.
Mohab Safey El Din is member of and supported by Institut Universitairede France.
We refer to [34] and [12] for basic definitions about algebraic sets and polynomial ideals. Givenan algebraic set V ⊂ C n , we denote by I ( V ) the ideal associated to V . Given f = ( f , . . . , f p ) in Q [ X , . . . , X n ], the set of their common solutions in C n is denoted by Z ( f ) and the ideal generatedby f is denoted by h f i . We say that f = ( f , . . . , f p ) is a reduced sequence when the ideal h f i generated by f is radical. Tangent spaces, regular and singular points.
Let V ⊂ C n be an algebraic set. For x ∈ V , thetangent space T x V at x to V is the vector space defined by P ni =1 ∂f∂X i ( x ) Y i = 0 for any f ∈ I ( V ).Also, given a finite set of generators f = ( f , . . . , f p ) of I ( V ), T x V is the kernel of the jacobianmatrix jac( f ) = (cid:16) ∂f i ∂X j (cid:17) ≤ i ≤ p ≤ i ≤ n . We denote by N x V the orthogonal complement to T x V .Assume now that V is d -equidimensional. The set of points x ∈ V where dim( T x V ) = d is the set ofregular points of V ; we denote it by reg( V ). The subset of singular points sing( V ) is the complementof reg( V ) in V ; it has dimension less than d . Observe that given a finite set of generators f of I ( V ),jac( f ) has rank n − d at all x ∈ reg( V ). Also, N x V is generated by the gradient vectors of thepolynomials in f evaluated at x . An equidimensional algebraic set V is said to be smooth whensing( V ) is empty. Zariski topology.
The Zariski topology over C n is the topology for which the closed sets are thealgebraic sets of C n . Let f ∈ C [ X , . . . , X n ]; we denote by O ( f ) ⊂ C n the subset defined by f = 0;it is a Zariski open set, which is non-empty when f is not identically 0. Further, we will provesome properties depending on parameters that are generically chosen. That means that, in theparameter space, there exists a non-empty Zariski open set such that the property is satisfied forany choice of the parameter values in this set. Projective varieties.
We will consider algebraic sets in the projective space P n ( C ) defined byhomogeneous polynomials. In the sequel, we use the shorthand P n for P n ( C ).Let V ⊂ P n be a projective variety. Notions of dimension, tangent space and regular (resp. singular)spaces extend to projective varieties. We denote by aff ( V ) ⊂ C n +1 the Zariski closure of the set4 ( x , . . . , x n ) ⊂ C n +1 | ( x : · · · : x n ) ∈ V } . The variety aff ( V ) is an affine cone (for all x ∈ aff ( V ), λ ∈ C we have λx ∈ aff ( V )). By a slight abuse of notation, when V is an algebraic set of C n , wealso denote by aff ( V ) the affine cone of the projective closure of V . Let now V ′ ⊂ C n +1 be an affinecone. Observe that the map proj : ( x , . . . , x n ) ∈ V ′ \ { } → ( x : · · · : x n ) ∈ P n sends V ′ \ { } to a projective set. Besides, for a projective variety V ⊂ P n , proj ( aff ( V )) = V . We also considerbi-projective varieties lying in P n × P n . The above constructions extend similarly: to any variety V ⊂ P n × P n can be associated a cone aff ( V ) ⊂ C n +1 × C n +1 which is the Zariski closure of theset of points ( x , . . . , x n , y , . . . , y n ) such that (( x : · · · : x n ) , ( y : · · · : y n )) ∈ V . The map proj isextended in the following way: proj : ( x, y ) ∈ ( C n +1 \ { } ) × ( C n +1 \ { } ) → ( x, y ) ∈ P n × P n . Let V ⊂ C n be a d -equidimensional algebraic set of codimension c and S ⊂ V be a subset. Followingthe terminology in [33, Chap. 5], an atlas for ( V, S ) is a finite sequence ψ = (( h j , m j )) ≤ j ≤ ℓ , with h j = ( h j, , . . . , h j,c ) ⊂ C [ X , . . . , X n ] and m j ∈ C [ X , . . . , X n ] such that for all 1 ≤ j ≤ ℓ thefollowing holds: P O ( m j ) ∩ ( V \ S ) = O ( m j ) ∩ ( Z ( h j ) \ S ); P O ( m j ) ∩ ( V \ S ) is not empty; P for all x ∈ O ( m j ) ∩ V \ S , jac( h j ) has full rank c at x ; P the open sets O ( m j ) cover V \ S .We say that h j is a set of local equations over O ( m j ). [33, Lemma 5.2.4] establishes that thereexists an atlas for ( V, sing( V )). Also, observe that sing( V ) ⊂ Z ( m · · · m ℓ ).Further, we use the notion of transverse intersection for algebraic sets and projective varieties. Let V and W be equidimensional algebraic sets in C n . As in [13, pp. 21], we say that V and W intersect transversely at x if x ∈ reg( V ) ∩ reg( W ) and T x V + T x W = C n . They intersect genericallytransversely if they meet transversely at a generic point of each irreducible component of V ∩ W .This definition is naturally extended to projective varieties.We say that two sets V and W intersect transversely over an open set U if V and W intersecttranversely at any point of V ∩ W ∩ U . Lemma 3.
Let V and V be equidimensional algebraic sets of codimensions c and c . Consideratlases α = (( h j , m j )) ≤ j ≤ ℓ and α = (( g j , n j )) ≤ j ≤ k for ( V , sing( V )) and ( V , sing( V )) . Assumethat V ∩ V is either empty or that for any irreducible component Z of V ∩ V , there exist r ∈{ , . . . , ℓ } and s ∈ { , . . . , k } such that T Z ∩ O ( m r n s ) is not empty; T At any point of reg( Z ) ∩ O ( m r n s ) , the matrix jac( h r , g s ) has rank c + c .Then V and V intersect generically transversely.Proof. The equality rank(jac( h r , g s )) = rank(jac( h r )) + rank(jac( g s )) implies that at any point x ∈ reg( Z ) ∩ O ( m r n s ), N x V ∩ N x V = 0. Consequently, T x V + T x V = ( N x V ∩ N x V ) ⊥ = C n .Finally, noticing that reg( Z ) ∩ O ( m r n s ) is dense in Z ∩ O ( m r n s ), which is dense in Z (by T )concludes the proof. 5e also need to prove that the intersection of bi-projective varieties is transverse. This is done viatheir associated affine cones. In the sequel, the set { ( x, y ) | x = } ⊂ C n +1 × C n +1 is denoted by X and the set { ( x, y ) | y = } ⊂ C n +1 × C n +1 is denoted by Y . Lemma 4.
Let V and V be projective varieties in P n × P n . Then V and V intersect transverselyat every point ( x, y ) = (( x : . . . : x n ) , ( y : . . . : y n )) ∈ P n × P n iff aff ( V ) and aff ( V ) intersecttransversely over C n +1 × C n +1 \ ( X ∪ Y ) .Proof. Let i, j be such that x i = 0 and y j = 0. W.l.o.g., we assume that i = j = 0. Consider theaffine chart U ⊂ P n × P n defined by x = 0 , y = 0. Let H ⊂ C n +1 × C n +1 (resp. H ) be thehyperplane defined by x = 1 (resp. y = 1). For ℓ ∈ { , } , the variety V ℓ ∩ U can be identifiedto aff ( V ℓ ) ∩ H ∩ H . By definition of transversality, the varieties V and V intersect transverselyat ( x, y ) ∈ P n × P n if and only if so do V ∩ U and V ∩ U . By the previous identification,this is equivalent to saying that aff ( V ) ∩ H ∩ H and aff ( V ) ∩ H ∩ H intersect transverselyat (1 , x , . . . , x n , , y , . . . , y n ). Finally, direct tangent space computations show that for z , z in C \ { } and for ℓ ∈ { , } , T ( z ,z x ,...,z x n ,z ,z y ,...,z y n ) aff ( V ℓ ) = T (1 ,x ,...,x n , ,y ,...y n ) aff ( V ℓ ). Let V ⊂ C n be an equidimensional algebraic set of codimension c and g ∈ Q [ X , . . . , X n ]. Con-sider the evaluation map ϕ g : x ∈ V → g ( x ). We denote by w ( ϕ g , V ) the set { x ∈ reg( V ) | rank(jac x ( f , g )) < c + 1 } . This is a locally closed constructible set and it coincides with the crit-ical locus of the map ϕ g . Its Zariski closure is denoted by W ( ϕ g , V ). This construction can begeneralized as follows.Let a , . . . , a n be linearly independent vectors in C n and for 1 ≤ i ≤ n , set a i = ( a , . . . , a i ) ∈ C i × n .Then, for 1 ≤ i ≤ n , let W (( g, a i ) , V ) denote the algebraic set { x ∈ V | rank(jac x ( f , g, ϕ a i )) < c + i + 1 } , where f is a set of generators of I ( V ), and ϕ a i is the set of linear forms ( a j · X ) j ∈{ ,...,i } (with X = ( X , . . . , X n )). Reusing the terminology of [19], we call these sets modified polar varieties associated to g and V , the i -th one being W (( g, a i ) , V ). We let W ( a i , V ) be the classical polarvariety { x ∈ V | rank(jac x ( f , ϕ a i ) < c + i } , reusing the letter W for the sake of simplicity. Proposition 5.
Let V be a d -equidimensional algebraic set, and i ∈ { , . . . , d } . There exists aZariski dense subset O ⊂ C i n and an integer integer numbers δ i such that for any a ∈ O , thefollowing holds: • W ( a, V ) is either empty or equidimensional of dimension i − ; • W ( a, V ) has degree at most δ i .Proof. The first statement follows directly from [1, Prop.3]. For the second statement, we refer tothe definition of δ classic in [4, Sec. 4].The integers δ i are denoted by δ i ( V ) in the sequel. By convention, we set δ d +1 = deg( V ). Thesenumbers are also called projective characters of V (see [16, Example 14.3.3]).6 Proof of Theorem 1
We start by introducing some objects which play a central role in the proof. As before, V is a d -equidimensional algebraic set and aff ( V ) denotes the affine cone over the projective closure of V .Let N V ⊂ C n +1 × C n +1 be the Zariski closure of the set { ( x, y ) ∈ C n +1 × C n +1 | x ∈ aff ( V ) \ { } , y ∈ N x aff ( V ) \ { }} . It is called the conormal variety of aff ( V ). Consider a = ( a , . . . , a i ) ∈ C ( n +1) i , a homogeneouspolynomial g ∈ C [ X , X , . . . , X n ] of degree D and the matrixΣ i ( g, a ) = Y · · · Y n ∂g/∂X · · · ∂g/∂X n a ... a i . Let S i ( g, a ) ⊂ C n +1 × C n +1 be the variety defined by the rank condition rank(Σ i ( g, a )) ≤ i + 1.Let Π be the projection Π : ( x, y ) ∈ C n +1 × C n +1 → x ∈ C n +1 . (3.1)If a is generic, then N V ∩ S i ( g, a ) is the Zariski closure of set of points ( x, y ) such that y ∈ N x aff ( V ),and ( y , . . . , y n ) ∈ Span( a ) + ∇ x g . In other words, ( x , x , . . . , x n ) is a critical point of the map( X , . . . , X n ) ( g ( X ) , a · X, . . . , a i · X ). Let a = ( a ′ , . . . , a ′ i − ) ∈ C n ( i − be a basis of the vectorspace { ( u , . . . , u n ) ∈ C n | (0 , u , . . . , u n ) ∈ Span( a , . . . , a i ) } . Therefore, if the first coordinate of a is nonzero, then the restriction of Π( N V ∩ S i ( g, a )) to the chart x = 1 is the modified polarvariety W (( g | x =1 , a ′ ) , V ). The set of homogeneous polynomials in C [ X , . . . , X n ] of degree D isa finite dimensional vector space; we denote by N its dimension and identify those homogeneouspolynomials to points in C N . Assume for the moment the following result which is proved inSection 4. Proposition 6.
Let V ⊂ C n be a d -equidimensional algebraic set such that its projective closure issmooth and i ∈ { , . . . , d } . There exists a non-empty Zariski open set O ⊂ C ( n +1) i × C N such thatfor any ( g, a ) ∈ O , N V and S i ( g, a ) meet generically transversely over ( C n +1 \ { } ) × ( C n +1 \ { } ) . One can associate to any equidimensional variety Z ⊂ P n × P n of codimension c a bivariate homoge-nous polynomial bideg( Z ) ∈ N [ T, U ] of degree c , called the bidegree of Z [36, 37]. The coefficient of T k U c − k in bideg( Z ) is the number of points (counted with multiplicity) of Z ∩ ( H × P n ) ∩ ( P n × H )where H (resp. H ) is a generic linear space of dimension n − k (resp. n − c + k ).By [10, Sec. 5], the bidegree of N V is P dk =0 δ k +1 ( V ) T n − k U k +1 .We focus now on the bidegree of S i ( g, a ). Lemma 7.
There exists a non-empty Zariski open set O ′ ⊂ C N × C ( n +1) i such that for ( g, a ) ∈ O ′ , S i ( g, a ) ⊂ C n +1 × C n +1 has codimension n − i and its bidegree is P n − ik =0 ( D − k T k U n − k − i . Moreover, reg( S i ( g, a )) coincides with the set of points where the matrix Σ i ( g, a ) has rank i + 1 . roof. S i ( g, a ) is the variety of ( x, y ) ∈ C n +1 × C n +1 where the evaluation of Σ i ( g, a ) is rankdefective. There exists a Zariski dense subset O ⊂ C ( n +1) i such that for all a ∈ O , the top-left i × i submatrix of A is invertible, where A is the matrix with rows a , . . . , a i . For a ∈ O , let B = ( b i,j ) be an invertible ( n + 1) × ( n + 1) matrix such that A · B = [ | I i ]. The rank conditionon A · B shows that S i ( g, a ) is the set of points ( x, y ) ∈ C n +1 × C n +1 where the rank of M = " P n +1 j =1 b j, Y j · · · P n +1 j =1 b j,n +1 − i Y j P n +1 j =1 b j, ∂g/∂X j · · · P n +1 j =1 b j,n +1 − i ∂g/∂X j is at most 1, where Y , . . . , Y n +1 − are new variables. Next, let S ′ i ⊂ P n − i × P n − i denote thedeterminantal variety of rank-defective matrices " u , · · · u ,n − i u , · · · u ,n − i , together with the grading given by deg( u ,j ) = 1, deg( u ,j ) = D − j ∈ { , . . . , n − i } .Setting s = 0, deg( t ) = D −
1, deg( t ) = 1, in [25, Example 15.39], the multidegree of S ′ i is P n − ik =0 ( D − k T k U n − k − i , where T (resp. U ) corresponds to the class of a hyperplane in the first(resp. second) operand in the product P n − × P n − . Let C [ X , . . . , X n ] D − denote the set ofhomogeneous polynomials of degree D −
1. Since determinantal varieties are Cohen-Macaulay[27, Thm. 11], by the same argument as in [15, Sec. 4], there exists a Zariski dense subset O ⊂ C [ X , . . . , X n ] n +1 D − × C ( n +1) i such that for any ( h , . . . , h n , a ) ∈ O , the bidegree of thevariety defined by rank( M ) ≤ P n − ik =0 ( D − k T k U n − k − i . Note that the set of( h , . . . , h n ) which are of the form ( ∂g/∂X , . . . , ∂g/∂X n ) is a linear subspace of C [ X , . . . , X n ]. Itremains to prove that the restriction of O to this subspace is nonempty. This is done by considering( h , . . . , h n ) = ( X D − , . . . , X D − n ) (which comes from the derivatives of g = ( X D + · · · + X Dn ) /D ) and a i,j = 1 if i = j and 0 otherwise. Direct computations show that the corresponding variety has theexpected bidegree. Therefore the open set O ′ of pairs ( g, a ) such that ( ∂g/∂X , . . . , ∂g/∂X n , a ) ∈ O satisfies the desired properties. Writing the equations defining the variety of S i ( g, a ) from the rankof the matrix M shows that ( x, y ) ∈ sing( S i ( g, a )) iff the evaluation of the first row of M is zero,which is equivalent to saying that ( Y , . . . , Y n ) lies in Span( a ). This implies that the regular locusof S i is the set of points where Σ i ( g, a ) has rank i + 1.By Proposition 6, there exists a non-empty Zariski open set O ⊂ C N × C ( n +1)( i +1) such that for g, a ′ in O , N V and S i +1 ( g, a ′ ) meet generically transversely outside the set X ∪ Y introduced beforeLemma 4. Consider the map proj introduced in Section 2 (paragraph on projective varieties).We deduce that for ( g, a ′ ) ∈ O , N ′ V = proj ( N V ) and S ′ i ( g, a ′ ) = proj ( S i ( g, a ′ )) meet genericallytransversely (Lemma 4). Below, we take ( g, a ) ∈ O ∩ O ′ (where O ′ is the non-empty Zariski openset defined in Lemma 7).Intersection theory [13, Theorem Definition 1.7] states that if two subvarieties Z and Z of P n × P n intersect generically transversely, thenbideg( Z ∩ Z ) = bideg( Z ) · bideg( Z ) mod h T n +1 , U n +1 i . We deduce that bideg( N ′ V ∩ S ′ i ( g, a )) equals d X k =0 δ k +1 ( V ) T n − k U k +1 ! n − i − X k =0 ( D − k T k U n − k − i − ! mod h T n +1 , U n +1 i . S ′ i +1 ( g, a ′ ) ∩ N ′ V by the projection π : ( x, y ) x is thecoefficient of T n − i − U n in its bidegree. Direct computations show that it equals ( δ i +1 ( V ) if d = 1 P dj = i δ j +1 ( V )( D − j − i otherwise . For j ∈ { , . . . , i + 1 } , let ν j be the first coefficient of a ′ j and let U be the set of a ′ ∈ C ( n +1)( i +1) such that ν = 0. Set ˜ O = { ( g, a ′ ) ∈ O ∩ O ′ | a ′ ∈ U } . For a ′ ∈ U , let χ be the map sending a ′ to ( a ′ − ν a ′ /ν , . . . , a ′ i − ν i a ′ /ν ) . The image of U by χ is a dense open subset U ′ ⊂ C ni . Finally,we write Ω for the set ( g | X =1 , a ) ∈ C [ X , . . . , X n ] ≤ D × C ni such that there exists ( g, a ′ ) ∈ ˜ O with χ ( a ′ ) = a . For ( g | X =1 , a ) ∈ Ω, S ′ i +1 ( g, a ′ ) and N ′ V intersect generically transversely. Moreover,its image by the projection Π (see (3.1)) restricted to the chart x = 1, y = 1 is W ( g | X =1 , a ).Consequently, deg( W ( g | X =1 , a )) ≤ deg(Π( S ′ i +1 ( g h , a ′ ) ∩ N ′ V ))= ( δ i +1 ( V ) if d = 1 P dj = i δ j +1 ( V )( D − j − i otherwise . Our proof relies on applying Lemma 3 with V = N V and V = S i ( g, a ) for a generic choice of ( a, g ).It simply consists in proving that properties T and T defined in Lemma 3 hold. This leads usto define atlases and local equations for N V . Next, we define an atlas (and hence local equations)for a set related to S i ( g, a ). We will apply an algebraic version of Thom’s weak transversalityTheorem to a well chosen map constructed using these local equations, establishing that this mapis regular at the origin. Finally, we will use these results in the last paragraph of this Section toprove properties T and T under some genericity assumption on ( g, a ). N V By assumption, V is d -equidimensional and smooth as is its projective closure; we denote by c its codimension. This implies that the affine cone aff ( V ) of the projective closure of V is alsoequidimensional of codimension c . Besides, if ( x, y ) ∈ aff ( V ) with x = 0 then x is a regular point of aff ( V ). By [33, Lemma 5.2.4], there exists an atlas ψ = (( h j , m j )) ≤ j ≤ J for ( aff ( V ) , sing( aff ( V )))(see Subsection 2.2). This leads us to define the set U j = { ( x, y ) | x ∈ aff ( V ) ∩ O ( m j ) , y ⊥ T x aff ( V ) \ { }} . Since the open sets O ( m j ) cover aff ( V ) \ sing( aff ( V )) (property P ), the sets U j cover N V \ X ∪ Y .Let m ′ j, , . . . , m ′ j,L j be the c × c minors of jac( h j ) such that O ( m j m ′ j,k ) ∩ V = ∅ for 1 ≤ k ≤ L j . For1 ≤ r ≤ n − c , we denote by M r,k ( m ′ j,k ) the minor of the ( c + 1 , c + 1) minors of the ( c + 1 , c + 1)submatrix of J = " jac( h j ) Y · · · Y n whose upper left ( c × c ) minor is m ′ j,k and adding the missing row and column. In the sequel, wedenote by H j,k the sequence h j , M ,k ( m ′ j,k ) , . . . , M n − c,k ( m ′ j,k ).9 emma 8. Under the above notation and assumptions, the sequence of couples ( H j,k , m j m ′ j,k ) for ≤ j ≤ J and ≤ k ≤ L j is an atlas for ( N V , sing( N V ) ∪ X ∪ Y ) .Proof. Recall that we are given an atlas ψ = (( h j , m j )) ≤ j ≤ J for ( aff ( V ) , sing( aff ( V ))). Let ( x, y ) ∈N V \ (sing( N V ) ∪ X ∪ Y ). Then, x ∈ reg( aff ( V )) (because x = and the projective closure of V is assumed to be smooth) and there exists 1 ≤ j ≤ J such that x ∈ aff ( V ) ∩ O ( m j ). Besides notethat aff ( V ) ∩ O ( m j ) coincides with Z ( h j ) ∩ O ( m j ) (property P ) and that jac( h j ) has maximalrank at x (property P ). We let m ′ j,k be a ( c × c )-minor of jac( h j ) which does not vanish at x .Since ( x, y ) ∈ N V , we have y ⊥ T x aff ( V ). Using property P and P , we deduce that T x aff ( V ) isthe kernel of jac( h j ). We deduce by elementary linear algebra that the matrix J introduced aboveis rank defective at ( x, y ). Besides, elementary linear algebra (e.g. using a Schur complement)shows that over O ( m j m ′ j,k ), the variety defined by h j ( x ) = 0 and rank ( J ( x, y )) ≤ c is defined by H j,k . We have established properties P and P . Establishing the fact that the sets O ( m j m ′ k )cover N V \ (sing( N V ) ∪ X ∪ Y ) (property P ) is immediate from the above discussion. It remainsto prove that jac( H j,k ) has maximal rank at ( x, y ) (property P ). Without loss of generality,assume that m ′ j,k is the upper left minor of jac( h j ). Observe that the minors M ,k ( m ′ j,k ) , . . . ,M n − c,k ( m ′ j,k ) can be written as Y c + ℓ m ′ j,k + ρ ℓ where ρ ℓ ⊂ Q [ X , . . . , X n , Y , . . . , Y c ]. Extractingfrom jac( H j,k ) the columns of jac( h j ) corresponding to m ′ j,k and those corresponding to the partialderivatives w.r.t Y c + ℓ for 1 ≤ ℓ ≤ n − c yields a submatrix which is not rank defective over Z ( h j ) ∩ O ( m j m ′ j,k ) which ends the proof. S i ( g, a ) In this section, we build an atlas for S i ( g, a ) for generic ( g, a ). To do that, we see ( g, a ) as inpoint in the space C N × C ( n +1) i (recall that N is the dimension of the vector space of homogeneouspolynomials in C [ X , . . . , X n ]) and see the entries of a and the coefficients of g as variables.Formally, for 1 ≤ r ≤ i , let A r = ( A ,r , . . . , A n,r ) be a vector of indeterminates. Let also M = { ( α , . . . , α n ) ∈ N n +1 | P nj =0 α j = D } and G = ( G α , α ∈ M ) be a vector of indeterminates. Byabuse of notation, we also denote by G the polynomial P α ∈M G α X α ; it lies in Q ( G )[ X , . . . , X n ].We consider now the matrix Σ i = Y · · · Y n ∂G/∂X · · · ∂G/∂X n A ... A i and the algebraic set S i ⊂ C n +1 × C n +1 × C N × C ( n +1) i defined by rank(Σ i ) ≤ i + 1.Let σ , . . . , σ L be the sequence of ( i + 1 , i + 1)-minors of the submatrix Σ i obtained by removingthe line containing partial derivatives of G or the line A j for 1 ≤ j ≤ L such that S i ∩ O ( σ ℓ ) = ∅ .For 1 ≤ ℓ ≤ L , we denote by S ,ℓ , . . . , S n − i − ,ℓ the ( i + 2 , i + 2)-minors of Σ i obtained by selectingthe rows and columns used to compute σ ℓ and adding the missing row and column from Σ i . Wedenote by S ℓ the sequence S ,ℓ , . . . , S n − i − ,ℓ .Finally, we define the set T ⊂ C N × C ( n +1) i as the complementary of the set of points ( g, a =( a , . . . , a i )) ∈ C N × C ( n +1) i such that • the coefficients of X r X D − s in G for 1 ≤ r, s ≤ n with r = s are not zero;10 ( g, a ) lies in the non-empty open set O defined in Lemma 7; • Span( a , . . . , a i ) has dimension i and none of the entries of A r is 0 (for 1 ≤ r ≤ i ).Note that T is Zariski closed in C N × C ( n +1) i . Finally, we denote by S ′ the union of sing( S i ), theset C n +1 × C n +1 × T and the subset of points S i such that their Y -coordinates are all 0.Up to renumbering the sequence of couples ( S ℓ , σ ℓ ) ≤ ℓ ≤ L we assume that the set of indices ℓ suchthat ( S i \ S ′ ) ∩ O ( σ ℓ ) = ∅ is { , . . . , L ′ } (for L ′ ≤ L ). Lemma 9.
The sequence ( S ℓ , σ ℓ ) ≤ ℓ ≤ L ′ is an atlas for the couple ( S i , S ′ ) . Besides, the truncatedJacobian matrix of S ℓ obtained by considering the partial derivatives w.r.t the entries of A , . . . , A i and the coefficients of G has full rank over O ( σ ℓ ) . Moreover, there exists a non-empty Zariski openset O ′′ such that for all ( g, a ) ∈ O ′′ , ( S ℓ , σ ℓ ) ≤ ℓ ≤ L ′ is an atlas of the couple ( S i ( g, a ) , sing( S i ( g, a )) .Proof. Take ( x, y, g, a ) in S i \ S ′ . Since ( g, a ) / ∈ S , ( g, a ) / ∈ O and ( x, y ) / ∈ sing( S i ( g, a )). Wededuce that Σ i has rank i + 1 at ( x, y, g, a ). Then, either dim(Span( a , . . . , a i , y )) = i + 1 or y ∈ Span( a , . . . , a i ) while ∇ x,y,a,g ( G ) / ∈ Span( a , . . . , a i ) (because Σ i has rank i + 1 at ( x, y, g, a )).Since ( y , . . . , y n ) = 0 (because ( x, y, g, a ) / ∈ S ), we deduce that there exists 1 ≤ r ≤ i such thatSpan( a , . . . , a r − , a r +1 , . . . , a i , y ) = Span( a , . . . , a i ) and we deduce thatdim(Span( a , . . . , a r − , a r +1 , . . . , a i , ∇ x ( g ) , y )) = i + 1 . This implies that one of the ( i + 1 , i + 1)-minor σ ℓ of Σ i does not vanish at ( x, y, g, a ). Elementarylinear algebra shows that S i ∩ O ( σ ℓ ) \ S ′ coincides with Z ( S ℓ ) over O ( σ ℓ ) \ S ′ . Thus, we haveestablished properties P and P . The covering property P is immediate and follows also from theabove discussion.It remains to prove property P , i.e. jac( S ℓ ) has maximal rank at any point of S i ∩ O ( σ ℓ ) \ S ′ .Assume first that σ ℓ is a ( i + 1 , i + 1)-minor obtained from removing the partial derivatives of G from Σ i . Without loss of generality, we may also assume that it is obtained by selecting the first i + 1 columns of Σ i . Then, polynomials in S ℓ can be written as σ ℓ A r,i +1 + ρ r,ℓ for i + 1 ≤ r ≤ n where ρ r,ℓ has degree 0 in A r . That implies that one can extract a diagonal matrix with σ ℓ on thediagonal from jac( S ℓ ) which, of course, has maximal rank over O ( σ ℓ ).When σ ℓ is obtained by removing one of the line A r (e.g. A i ) a more involved but similar conclusioncan be made. Since we work over the complementary of S ′ , there exists 0 ≤ r ≤ n such that the X r -coordinate of x is not 0. Extracting the submatrix of jac( S ℓ ) corresponding to the partial derivativeswith respect to the coefficients of G of the monomials X Dr and X s X D − r yields a diagonal matrixwith a power of the X r -coordinate of x multiplied by σ ℓ on the diagonal. These are non-zero over O ( σ ℓ ) \ S ′ .The rank property of the truncated Jacobian matrix of S ℓ is an immediate consequence of theabove discussion. Details on the proof of the specialization property of the atlas ( S ℓ , σ ℓ ) are left tothe reader; we mention that it is a direct consequence of specialization properties of minors withpolynomial entries and Lemma 7. Let m j , m ′ k and H j,k be the polynomials introduced in the paragraph on local equations for N V and σ ℓ , S ℓ be the ( i + 1 , i + 1)-minor and ( i + 2 , i + 2)-minors of Σ introduced in the paragraph onlocal equations of S i ( g, a ). Consider the Zariski open set U j,k,ℓ ⊂ C n × C n × C N × C ni defined by m j m ′ k = 0 , ( X , . . . , X n ) = ( Y , . . . , Y n ) = , σ ℓ = 011nd the inequations defining the complement S ′ . We define now the following map: φ j,k,ℓ : z ∈ U j,k,ℓ → ( H j,k ( z ) , S ℓ ( z )) . Observe that φ − j,k,ℓ ( ) ⊂ N V ∩ S i . Lemma 10.
The map φ j,k,ℓ is regular at .Proof. Since j, k and ℓ are fixed in the sequel, we omit them as subscripts. Observe that jac( H , S )has the following shape J φ = " jac X ( H ) jac X ( S ) ∆ where the last columns correspond to the partial derivates with respect to the entries of A , . . . , A i and G . By Lemma 8, ( H j,k , m j m ′ k ) satisfies properties P , P and P . This implies that it hasmaximal rank at any point in φ − ( ) ⊂ U . By Lemma 9, ∆ has maximal rank at any pointof φ − ( ). We deduce that J φ has maximal rank at any point of φ − ( ) and our conclusionfollows.In the sequel, for ( g, a ) ∈ C N × C ( n +1) i , we denote by φ ( g,a ) j,k,ℓ the restricted map ( x, y ) → φ j,k,ℓ ( x, y, g, a ).Applying Thom’s weak transversality Theorem (see [33, Sec 4.2]) to φ j,k,ℓ shows that there existsa non-empty Zariski open set O ′′′ j,k,ℓ ⊂ C N × C ( n +1) i such that for all ( g, a ) ∈ O ′′′ j,k,ℓ , the restrictedmap φ ( g,a ) j,k,ℓ is regular at . Letting O ′′′ be the intersection of all these non-empty Zariski open sets O ′′′ j,k,ℓ leads to the following result. Lemma 11.
There exists a non-empty Zariski open set O ′′′ ⊂ C N × C ( n +1) i such that for any ( j, k, ℓ ) and ( g, a ) ∈ O ′ , the restricted map φ ( g,a ) j,k,ℓ is regular at . Let O be the intersection of the non-empty Zariski open sets O ′′ and O ′′′ defined in Lemma 9and Lemma 11. Take ( g, a ) ∈ O and Z ( g,a ) be the Zariski closure of S j,k,ℓ φ ( g,a ) j,k,ℓ − ( ). Recallthat we need to prove the transversality of N V ∩ S i ( g, a ) at any point outside X ∪ Y . Let α = ( H j,k , m j m ′ k ) be the atlas of ( N V , sing( N V )) defined in Lemma 8 and α = ( S ℓ , σ ℓ ) be theatlas of ( S i ( g, a ) , sing( S i ( g, a ))) defined in Lemma 9. We start by proving that the Zariski closureof N V ∩ S i ( g, a ) \ ( X ∪ Y ) equals Z g,a . The inclusion Z ( g,a ) ⊂ N V ∩ S i ( g, a ) \ ( X ∪ Y ) is immediatesince all points in φ ( g,a ) j,k,ℓ − ( ) ⊂ Z ( H j,k , S k,ℓ ) ∩ O ( m j m ′ k σ ℓ ) and Z ( H j,k , S k,ℓ ) ∩ O ( m j m ′ k σ ℓ ) = N V ∩ S i ( g, a ) ∩ O ( m j m ′ k σ ℓ ) (property P ). We prove now the reverse inclusion. It is sufficient toprove that for any irreducible component Z of the Zariski closure of N V ∩ S i ( g, a ) \ ( X ∪ Y ), thereexists a triple ( j, k, ℓ ) and a Zariski closed subset F ( Z such that Z \ F ⊂ φ ( g,a ) j,k,ℓ − ( ). Since Z is anirreducible component of the Zariski closure of N V ∩ S i ( g, a ) \ ( X ∪ Y ), there exists ( x, y ) ∈ Z suchthat ( x, y ) / ∈ X ∪ Y . Let F = Z ∩ ( X ∪ Y ). Now, take ( x, y ) ∈ Z \ F . By property P applied to α ,that implies that there exists j and k such that x ∈ Z ( H j,k ) ∩ O ( m j m ′ k ). Besides, ( y , . . . , y n ) = since ( x, y ) / ∈ F . This latter property implies that there exists ℓ such that σ ℓ ( x, y ) = 0. Finally, wehave established that ( Z \ F ) ∩ O ( m j m ′ k σ ℓ ) is not empty for some ( j, k, ℓ ). Property P applied to α and α imply that ( x, y ) lies in Z ( H j,k ) and Z ( S ℓ ). We deduce that ( x, y ) ∈ φ ( g,a ) j,k,ℓ − ( ) whichimplies that Z \ F ⊂ φ ( g,a ) j,k,ℓ − ( ) as requested. 12 roperty ( T ) . Consider an irreducible component Z of the Zariski closure of N V ∩ S i ( g, a ) \ ( X ∪ Y ). The above discussion implies that Z is an irreducible component of Z ( g,a ) and that thereexists j, k, ℓ such that Z ∩ O ( m j m ′ k σ ℓ ) is not empty. Property ( T ) . Recall that ( g, a ) ∈ O ′ and let Z be an irreducible component of N V ∩ S i ( g, a ).We already proved that Z there exists ( j, k, ℓ ) such that Z ∩ O ( m j m ′ k σ ℓ ) is not empty and meets φ ( g,a ) j,k,ℓ − ( ). By Lemma 11, the restricted map φ ( g,a ) j,k,ℓ is regular at . Then, the jacobian matrixassociated to H j,k , S ℓ has maximal rank at any point of Z ∩ φ ( g,a ) j,k,ℓ − ( ), which concludes the proof. We show in this section that the bounds in Theorem 1 hold under milder conditions than thegenericity of the coefficients of g . Consider a d -equidimensional algebraic set V ⊂ C n whoseprojective closure is smooth, a set of generators f , . . . , f p of I ( V ), and g ∈ Q [ X , . . . , X n ] of degree D . Let a ∈ C ni , g ∈ Q [ X , . . . , X n ] and I crit ( g, a ) be the ideal generated by f , . . . , f p and the( n − d + i + 1)-minors of the matrix jac( f ) ∂g/∂X · · · ∂g/∂X n a ... a i . Proposition 12.
Let i ∈ { , . . . , d } and a ∈ C ni . Assume that the ideal I crit ( g, a ) is radical and W ( g, a ) is empty or ( i − equidimensional. Then there exists a non-empty Zariski open subset O ⊂ C in such that the following holds. For any a = ( a , . . . , a i ) ∈ O , the degree of W (( g, a ) , V ) isbounded above by the bounds in Theorem 1. Lemma 13.
Let Q ∈ C [ T , . . . , T ℓ ] be a nonzero multivariate polynomial, and ( t , . . . , t ℓ ) ∈ C ℓ besuch that Q ( t , . . . , t ℓ ) = 0 . Then there exist univariate polynomials u , . . . , u ℓ ∈ C [ e ] such that forall i ∈ { , . . . , ℓ } , u i (0) = t i and Q ( u ( e ) , . . . , u ℓ ( e )) ∈ C [ e ] is not identically zero.Proof. We prove the existence of u , . . . , u ℓ of the form u i ( e ) = t i + s i e , where s i ∈ C for all i ∈ { , . . . , ℓ } . Let t and s be shorthands for ( t , . . . , t ℓ ) and ( s , . . . , s ℓ ). Using Taylor’s expansion,we write Q ( t + e s ) = e ∂Q ( t )( s ) + e ∂ Q ( t )( s , s ) / . . . + e deg( Q ) ∂ deg( Q ) Q ( t )( s , . . . , s ) / deg( Q )!.Since Q = 0, at least one of its derivatives is not zero at t . Let k be the smallest integer suchthat u ∂ k Q ( t )( u , . . . , u ) is not the zero map. Finally, let s be such that ∂ k Q ( t )( s , . . . , s ) = 0.Hence, we have Q ( t + e s ) − e k ∂ k Q ( t )( s , . . . , s ) /k ! = O ( | e k +1 | ). Consequently, Q ( t + e s ) cannot beidentically zero, as this would imply e k = O ( | e k +1 | ). of Proposition 12. The proof is a classical deformation argument similar to the one used in [26].Further we assume that W ( g, a ) is not empty (else the result is immediate). By Theorem 1, thereexists a polynomial Q in N + ni = (cid:0) n + Dn (cid:1) + ni variables whose zero-set encode the pairs ( g, a ) forwhich the bounds are not satisfied. By Lemma 13, there exists ( g , a ) ∈ Q [ e ][ X , . . . , X n ] × Q [ e ] ni such that their evaluation at e = 0 is ( g, a ) and the evaluation of Q at ( g , a ) (seen as an elementin Q [ e ] N + ni ) is nonzero. For ε ∈ C , we let ( g ε , a ε ) denote the evaluation of g and a at e = ε . For i ∈ { , . . . , d } , the set of affine spaces in C n of codimension i − n − i + 2)-dimensional vector spaces in C n +1 . Since W (( g, a ) , V ) is( i − O of this Grassmanian such that for any E in O , the intersection W (( g, a ) , V ) ∩ E is transverse, finite and its cardinality equals the degreeof W (( g, a i ) , V ). Let x ∈ C n be a point in this intersection. Let v , . . . , v n ∈ C [ X , . . . , X n , e ] bepolynomials satisfying the following assumptions: v , . . . , v n is a regular sequence, for every ε ∈ C their evaluations at e = ε vanish on W (( g ε , a ε ) , V ) ∩ E , and the jacobian matrix jac( v ( X , . . . , X n , , . . . , v n ( X , . . . , X n , x (since I crit ( g, a ) is radical). In order to obtain suchpolynomials, we consider n generic linear combinations of the equations defining W (( g , a ) , V ) ∩ E .Then the holomorphic implicit mapping theorem [24, Thm. 8.6] states that for x ∈ W (( g, a ) , V ) ∩ E there exist open neighborhoods (for the Euclidean topology) 0 ∈ U ⊂ C , x ∈ U such that thereis a holomorphic map ε
7→ { x ∈ U | v ( x , ε ) = · · · = v n ( x , ε ) = 0 } on U . In particular this map iscontinuous, which implies that for ε ∈ C with sufficiently small complex modulus, the cardinalityof W (( g ε , a ε ) , V ) ∩ E is bounded below by the degree of W (( g, a ) , V ). Since this is true for any E in the Zariski dense open subset O of affine subsets, the cardinality of W (( g ε , a ε ) , V ) ∩ E equalsits degree. Finally, as Q is not identically zero on the coefficients of ( g , a ), for ε with sufficientlysmall modulus, the evaluation of Q at the coefficients of ( g ε , a ε ) is nonzero. Consequently, thebounds in Theorem 1 hold for W (( g ε , a ε ) , V ) and hence they also hold for W (( g, a ) , V ). Terminology and computational model.
In this section, we consider bounded error probabilis-tic algorithms . These algorithms are probabilistic random-access stored-program machines whoseprobability of success is bounded from above by an a priori bound. It is the same computationalmodel as in [17]. Complexity bounds count the number of arithmetic operations (+, − , × , / ) in Q .A lifting fiber is a data structure giving an exact representation of an equidimensional algebraicset. We recall below its definition and we refer to [17, Sec. 3.4] for more details. Definition 14. [17, Def. 4] Let V ⊂ C n be a d -equidimensional variety defined over Q ( i.e. Z C ( I Q ( V )) = V ). A lifting fiber for V is a tuple L = ( G , M, z , u, Q, v ) : • A lifting system H = ( h , . . . , h n − d ) ∈ Q [ X , . . . , X n ] , such that h , . . . , h n − d is a reducedregular sequence and V ⊂ Z ( H ) . • A n × n invertible matrix M with entries in Q such that the coordinates Y = M − X are in Noether position w.r.t. V ; • A rational lifting point z = ( z , . . . , z d ) ∈ Q d ; • A primitive element u : C n → C , which is a linear form with rational coefficients havingdistinct values at all points of the finite set V ( z ) = V ∩ { Y − z = · · · = Y d − z d = 0 } ⊂ C n ; • A polynomial Q ∈ Q [ T ] of minimal degree vanishing at all points of u ( V ( z ) ) ; • univariate polynomials v = ( v d +1 , . . . , v n ) ∈ Q [ T ] n − d of degree less than deg( Q ) such that Y − z = · · · = Y d − z d = 0 Y d +1 − v d +1 ( T ) = · · · = Y n − v n ( T ) = 0 , Q ( T ) = 0 is a rational parametrization of V ( z ) by the roots of Q . he sequence ( M, u, Q, v ) is called a geometric resolution of V . Computing a lifting fiber can be achieved in a probabilistic way with the Kronecker solver [11]. Weassume that we know a probabilistic algorithm
PolarVar which takes as input d ∈ N , a liftingfiber of a d -equidimensional variety V ⊂ C n and a sequence a = ( a , . . . , a d ) ∈ Q d × n ; it returns ageometric resolution of the 0-dimensional polar variety W ( a , V ) or “fail”. We use also the routine ChangePrimitiveElement [17, Algo. 6].In [1], the authors propose an algorithm which takes as input a reduced regular sequence f , . . . , f n − d defining a d -equidimensional algebraic set V ⊂ C n , a matrix F whose entries are multivariatepolynomials, and a sequence a = ( a . . . , a d ) of vectors in Q n . It returns lifting fibers for theassociated degeneracy loci . If F turns out to be the jacobian matrix of the regular sequence definingthe variety, then these degeneracy loci are the classical polar varieties W ( a, V ), see [1, Section 5.1].This algorithm works in two steps: it computes first a lifting fiber for V ; then, from this liftingfiber and from the matrix a , it computes lifting fibers for the degeneracy loci. In the case of polarvarieties, the complexity of the second step is bounded by L ( nD max ) O (1) δ , where δ is the maximumof the degrees of the polar varieties W ( a i , V ) (where a i = ( a , . . . , a i )), D max is the maximum ofthe degrees of f , . . . , f n − d , and L is the size of an essentially division-free straight line program forevaluating f , . . . , f n − d .Let a = ( a , . . . , a d ) be a sequence of d vectors in Q n . We construct another sequence a ′ =( e n +1 , a ′ , . . . , a ′ d ) of vectors in Q n +1 defined by the ( d + 1) × ( n + 1) coefficient matrix A ′ = a ... ... ... a d . Lemma 15.
Let Π n : C n +1 → C n be the projection on the n first coordinates, and f , . . . , f p ∈ Q [ X , . . . , X n ] be polynomials defining a reduced smooth d -equidimensional variety and g ∈ Q [ X , . . . , X n ] be a polynomial. Then for any a ∈ Q d × n and for i ∈ { , . . . , d } , the modified polar variety W (( g, a ) , Z ( f , . . . , f p )) equals Π n ( W ( a ′ i +1 , Z ( f , . . . , f p , g − X n +1 ))) . Proof.
Set V = Z ( f , . . . , f p ) ⊂ C n and V ′ = Z ( f , . . . , f p , g − X n +1 ) ⊂ C n +1 . Direct computationsshow that if V is smooth, then so is V ′ . The modified polar variety W (( g, a i ) , V ) is defined by theset of points in V at whichrank jac( f ) ∇ ga ... ... ... a ≤ n − d + i. Direct computations show that the corresponding matrix for W ( a ′ i , V ′ ) has the same rank at anypoint ( x, g ( x )) where x ∈ V . 15 lgorithm 1: CritPoints
Input : • A lifting fiber ( H , M, z , u, Q, v ) for a smooth d -equidimensional variety V ⊂ C n • g ∈ Q [ X , . . . , X n ] and a = ( a , . . . , a d ) ∈ Q d × n • A primitive element u crit for W ( a , V ) M ′ ← M ; L ′ ← ( H ∪ { g − X n +1 } , M ′ , ( z , . . . , z n , g ( z , . . . , z n )) ,u, Q, ( v , . . . , v n , g ◦ ( v ( T ) , . . . , v n ( T )) mod Q ( T ))); a ′ ← sequence of rows of a ... ... ... a d ; L (2) ← PolarVar ( d, L ′ , a ′ ) or return “fail”;( H ′ , M (2) , z ′ , u crit , Q ′ , v ′ ) ← ChangePrimitiveElement ( L (2) , u crit ◦ M (2) ); v (2) ← ( M (2) ) − · ( v ′ , . . . , v ′ n +1 ) T ;return ( Id n , u crit ◦ ( M (2) ) − , Q ′ , ( v ′ , . . . , v ′ n )); 16 heorem 16. Let L = ( H , M, z , u, Q, v ) be a lifting fiber for a d -equidimensional algebraic set V , g ∈ Q [ X , . . . , X n ] be a polynomial of degree D ≥ and a ∈ C ni . Assume that V, g and a satisfy the same assumptions as in Proposition 12. Let D max be the maximum of the degreesof h , . . . , h n − d , g , (where H = ( h , . . . , h n − d ) ) and u be a primitive element for W ( g, V ) . Assumethat the evaluation map x ( h ( x ) , . . . , h n − d ( x ) , g ( x )) is represented by an essentially division-freestraight-line program of size L . Algorithm 1 with input ( L , a , u ) computes a geometric resolutionof the set W ( g, V ) or it returns “fail”. Using the algorithm in [1] for PolarVar , it requires atmost ( nD max ) O (1) e O ( L ∆ ) operations in Q , where ∆ = P dj =0 δ j +1 ( V )( D − j − i .Proof. We prove first the correctness of the algorithm. Note that L ′ computed during Algorithm1 is a lifting fiber for V ′ = { ( x, g ( x )) | x ∈ V } . Assuming that PolarVar returns a lifting fiber for W ( a ′ , V ′ ), Lemma 15 shows that the output of PolarVar is a lifting fiber of the pairs ( x, g ( x )) for x ∈ W ( g, V ). The last steps compute a geometric resolution of the projection on the n first coor-dinates, which is W ( g, V ). We prove now the complexity statement. The first step of Algorithm 1does not cost any arithmetic operations. The second step requires e O ( L deg( V )) operations in Q for the modular composition using quasi-linear algorithms for multiplication and reduction. Theevaluation of g costs L operations. The cost of the computation of a ′ is negligible. By [1, Thm. 18],the call to PolarVar requires L ( pnd ) O (1) δ ′ , where δ ′ is the maximum of the degrees of the polarvarieties of V ′ . By Lemma 15, the projection of W ( a ′ , V ′ ) on the n first coordinates is W ( g, V ).By Theorem 1, Proposition 12 and since deg( g ) ≥
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