Smooth Points on Semi-algebraic Sets
SSmooth Points on Semi-algebraic Sets
Katherine Harris a , Jonathan D. Hauenstein b , Agnes Szanto a a Department of Mathematics, North Carolina State University,Campus Box 8205, Raleigh, North Carolina, 27695, USA. b Department of Applied and Computational Mathematics and Statistics, University of Notre Dame,102G Crowley Hall, Notre Dame, Indiana, 46556, USA.
Abstract
Many algorithms for determining properties of real algebraic or semi-algebraic sets rely upon theability to compute smooth points. In this paper, we present a simple procedure based on comput-ing the critical points of some well-chosen function that guarantees the computation of smoothpoints in each connected bounded component of a real atomic semi-algebraic set. Our techniqueis intuitive in principal, performs well on previously di ffi cult examples, and is straightforward toimplement using existing numerical algebraic geometry software. The practical e ffi ciency of ourapproach is demonstrated by solving a conjecture on the number of equilibria of the Kuramotomodel for the n = ffi cient algorithm to com-pute the real dimension of algebraic sets, the original motivation for this research. We comparethe e ffi ciency of our method to existing methods to compute the real dimension on a family ofbenchmark problems. Keywords: computational real algebraic geometry, real smooth points, real dimension, polarvarieties, numerical algebraic geometry, Kuramoto model
1. Introduction
Consider the atomic semi-algebraic set S = { x ∈ R n : f ( x ) = · · · = f s ( x ) = , q ( x ) > , . . . , q m ( x ) > } (1)for some f , . . . , f s , q , . . . , q m ∈ R [ x , . . . , x n ]. When studying semi-algebraic sets, one oftenfirst studies the complex variety V = { x ∈ C n : f ( x ) = · · · = f s ( x ) = } and deduces propertiesof S from the properties of V . In particular, if S contains a smooth point and V is irreducible,then S is Zariski dense in V , so all of the algebraic information of S is contained in V . Thus,deciding the existence of smooth points in real semi-algebraic sets and finding such points is acentral problem in real algebraic geometry with many applications. For example, if ϕ : S → S (cid:48) is a polynomial map of semi-algebraic sets, then smooth points in Im( ϕ ) are points where theJacobian of ϕ has maximal rank within its connected component, called the typical rank . Finding Email addresses: [email protected] (Katherine Harris), [email protected] (Jonathan D. Hauenstein), [email protected] (Agnes Szanto)
Preprint submitted to Journal of Symbolic Computation July 17, 2020 a r X i v : . [ c s . S C ] J u l eal smooth points in each connected component of a semi-algebraic set allows one to computeall typical ranks of real morphisms (see Sottile (2019) for applications of this property).One of the main results of this paper is to give a new technique to compute smooth pointson bounded connected components of atomic semi-algebraic sets. Our method is simple andsuggests natural implementation using numerical homotopy methods and deformations. It com-plements other approaches that compute sample points on real semi-algebraic sets, such as com-puting the critical points of the distance function, in the sense that our method also guaranteesthe smoothness of the sample points. We demonstrate this advantage on “Thom’s lips” in whichcritical points of the distance function are often at the singularities (Wu and Reid, 2013, Ex. 2.3),while our method always computes smooth points. The main idea is very simple. Suppose V is irreducible. If a polynomial g vanishes on the singular points of V , but does not vanish onall of V , then the extreme points of g on S must contain nonsingular points in every boundedconnected component of S , if such points exist. We extend this idea to the case when V is notequidimensional (i.e. reducible and the components may have di ff erent dimensions) by using in-finitesimal deformations of V and limits. We show that this limiting approach is well-suited fornumerical homotopy continuation methods after we translate an infinitesimal real deformation(that may only work for arbitrary small values) into a complex deformation that works alonga real arc parameterized by the interval (0 , g using deflations, and compare its degree bounds to traditional symbolicapproaches (see Proposition 4.8). In fact, Corollary 4.9 proves that our R eal S mooth P oint A l - gorithm performs well if the depth of the deflations (i.e. the number of iterations) s small.To demonstrate the practical e ffi ciency of our new approach, we present the solution of aconjecture for the first time: counting the equilibria of the Kuramoto model in the n = ffi -culty of this problem, compared to its complex counterpart, is that in many cases the real part lieswithin the singular set of the complex variety containing it, and its real dimension is smaller thanthe complex one. In terms of worst case complexity bounds of the existing algorithms in the lit-erature, it is an open problem if the real dimension can be computed within the same asymptoticcomplexity bounds as the complex dimension. The motivation for this research was to try to findan algorithm for the real dimension that has worst case complexity comparable to its complexcounterpart. Even though this paper is presented using computational tools from numerical alge-braic geometry (c.f. Sommese and Wampler (2005); Bates et al. (2013)), all procedures can betranslated to symbolic methods for polynomials with rational coe ffi cients. In fact, we did a worstcase complexity estimate for a symbolic version, and found that unfortunately it does not improvethe existing complexity bounds in the worst case (see Bannwarth and Safey El Din (2015) andthe references therein). This is one of the reasons we wrote the paper in a numerical algebraicgeometry setting, and gave evidence of the e ffi ciency on benchmark problems. As mentionedabove, in Proposition 4.8 and Corollary 4.9, we give bounds on the degrees of the polynomialsappearing in our algorithms and the number of homotopy paths they follow, highlighting theadvantages and disadvantages of our approach compared to other purely symbolic techniques. There are many approaches in the literature to compute at least one real point on every con-nected component of a semi-algebraic set. Methods using projections to obtain a cell decompo-sition based on sign conditions go back to Collins’ Cylindrical Algebraic Decomposition (CAD)2lgorithm described in Collins (1975). Improved symbolic methods using critical points or gen-eralized critical points of functions along with infinitesimals and randomization can be found inRouillier et al. (2000); Aubry et al. (2002); Safey El Din (2007); Faug`ere et al. (2008). The cur-rent state of the art deterministic symbolic algorithm is given in (Basu et al., 2006a, Alg. 13.3)which computes sample points on each connected component of all realizable sign conditionsof a polynomial system and gives a complexity analysis. The most recent application of thistechnique is in Safey El Din et al. (2018, 2019) where the authors compute smooth points on realalgebraic sets in order to compute the real radical of polynomial systems and analyze complexity.Alternatively, a homotopy-based approach computing the critical points of the distance functionfrom a generic point or a line is presented in Hauenstein (2013); Wu and Reid (2013).Another line of work has been developed in parallel which specifically focuses on computingcritical points while utilizing the tool of polar varieties , introduced and developed in Bank et al.(1997); Safey El Din and Schost (2003); Bank et al. (2004, 2009, 2010, 2015); Safey El Din andSpaenlehauer (2016). It is important to note, however, that all of these methods only guaranteethe finding of real points on every connected component of a semi-algebraic set, rather than real smooth points.The real dimension problem has similarly been widely studied with the current state of the artdeterministic algorithm is given by (Basu et al., 2006a, Alg. 14.10) computing all realizable signconditions of a polynomial system. This approach improves on previous work in Vorobjov (1999)to obtain a complexity result with a better dependence on the number of polynomials in theinput by utilizing a block elimination technique first proposed in Grigor (cid:48) ev and Vorobjov (1988).Recent work has been presented giving probabilistic algorithms utilizing polar varieties whichimprove on complexity bounds even further in Safey El Din and Tsigaridas (2013); Bannwarthand Safey El Din (2015). We use a benchmark family from Bannwarth and Safey El Din (2015)to demonstrate the e ffi ciency of our method.One can also compute the real dimension by computing the real radical of a semi-algebraicset, first studied in Becker and Neuhaus (1993) with improvements and implementations inNeuhaus (1998); Zeng (1999); Spang (2008); Chen et al. (2013). The most recent implemen-tation can be found in Safey El Din et al. (2018, 2019) as mentioned above. Their approach isshown to be e ffi cient in the case when the polynomial system is smooth, but the iterative com-putation of singularities of singularities can increase the complexity significantly in the worstcase. An alternative method using semidefinite programming techniques was proposed by Wang(2016); Ma et al. (2016).
2. Preliminaries
The following collects some basic notions used throughout including atomic semi-algebraicsets, semi-algebraic sets, and real algebraic sets.A set S ⊂ R n is an atomic semi-algebraic set if it is of the form of (1). A set T ⊂ R n is a semi-algebraic set if it is a finite union of atomic semi-algebraic sets. A set U ⊂ R n is a realalgebraic set if it is defined by polynomial equations only.Smoothness on atomic semi-algebraic sets is described next. Definition 2.1.
Let S ⊂ R n be an atomic semi-algebraic set as in (1). A point z ∈ S is smooth(or nonsingular) in S if z is smooth in the algebraic set V ( f , . . . , f s ) = { x ∈ C n : f ( x ) = · · · = f s ( x ) = } , V ⊂ V ( f , . . . , f s ) containing z such thatdim T z ( V ) = dim V where T z ( V ) is the tangent space of V at z . We denote by Sing( S ) the set of singular (or non-smooth) points in S .An algebraic set V ⊂ C n is equidimensional of dimension d if every irreducible componentof V has dimension d . The following defines the real dimension of semi-algebraic sets from(Basu et al., 2006a, § Definition 2.2.
For a semi-algebraic set S ⊂ R n , its real dimension dim R S is the largest k suchthat there exists an injective semi-algebraic map from (0 , k to S . Here, a map ϕ : (0 , k → S is semi-algebraic if the graph of ϕ in R n + k is semi-algebraic. By convention, the dimension (realor complex) of the empty set is − Theorem 2.3.
Let V ⊂ C n be an irreducible algebraic set and let V R : = V ∩ R n . Then dim R V R = dim C Vif and only if there exists z ∈ V R that is smooth.2.1. Semi-algebraic to Algebraic In this section, we show that our problem on atomic semi-algebraic sets can be reformulatedas a problem on real algebraic sets. This will allow us to use homotopy continuation methodsthat solve polynomial equations, but not inequalities.The following shows that smooth points on each connected component of an atomic semi-algebraic set S can be obtained as projections of smooth points of some real algebraic set. Proposition 2.4.
Let S be an atomic semi-algebraic set as in (1) andW : = (cid:110) ( x , z ) ∈ R n × R m : f ( x ) = · · · = f n ( x ) = , z q ( x ) − = · · · = z m q m ( x ) − = (cid:111) . If y ∈ W is smooth, then π x ( y ) ∈ S is also smooth. Conversely, if x ∈ S is smooth, then ( x , z ) issmooth in W for all z = ( z , . . . , z m ) ∈ R m such that ( x , z ) ∈ W.Proof.
Without loss of generality, we can assume that f , . . . , f s generate a prime ideal. TheJacobian matrix of the polynomial system defining W has the block structure J ( x , z ) = J f ( x ) 0 ∗ diag(2 z i q i ( x ))Since for ( x , z ) ∈ W we have z i g i ( x ) (cid:44)
0, the Jacobian matrix
J f ( x ) has full column rank if andonly if J ( x , z ) has full column rank, which proves the claim.Therefore, for the rest of the paper, we assume that we are given a real algebraic set and thegoal is to compute smooth points on each connected component.4 .2. Boundedness The next reduction is to replace an arbitrary real algebraic set with a compact one.
Proposition 2.5.
Let f , . . . , f s − ∈ R [ x , . . . , x n − ] and consider q = ( q , . . . , q n − ) ∈ R n − . Let δ ∈ R + , introduce a new variable x n , and considerf s : = ( x − q ) + · · · + ( x n − − q n − ) + x n − δ Then, V ( f , . . . , f s ) ∩ R n is bounded and π n − ( V ( f , . . . , f s ) ∩ R n ) = V ( f , . . . , f s − ) ∩ (cid:110) z ∈ R n − : (cid:107) z − q (cid:107) ≤ δ (cid:111) where π n − ( x , . . . , x n ) = ( x , . . . , x n − ) . Remark 2.6.
The definition of f s above is based on a standard trick used in real algebraic ge-ometry to make an arbitrary real algebraic set bounded (e.g., see Basu et al. (2006b)). In general, V ∩ R n − is embedded into a sphere in R n around the origin of radius 1 /ζ where ζ is infinitesimal.In this paper, we are only interested in computing points with bounded coordinates, so it is suf-ficient to embed its intersection with a closed ball around q of radius √ δ for some fixed δ ∈ R + .In particular, we will not use infinitesimal variables.Later in the paper, when we assume that V ( f , . . . , f s ) ∩ R n is compact, we assume that weapplied Proposition 2.5. The algorithms described in this paper make assumptions that certain points, matrices, orlinear polynomials are generically chosen from a vector space (over Q , R or C ). In all thesecases, there exists a proper Zariski closed subset of the corresponding vector space such that allchoices outside this set yield correct answers. Therefore, a generic choice means it is outsideof this proper Zariski closed subset. For algorithms which depend on generic choices, the al-gorithms compute the correct answer with algebraic probability one (Sommese and Wampler,2005, Chap. 4). E ff ective probability bounds can be obtained from bounds on the degrees of theproper Zariski closed sets containing the “bad” choices. See (Krick et al., 2001, Prop. 4.5) andElliott and Schost (2019) for such bounds for linear changes of variables for Noetherian positionand transversality, respectively.
3. Computation of Real Smooth Points
This section contains the main algorithms of this paper with the subsequent section providingthe necessary subroutines.
Theorem 3.1.
Let f , . . . , f s ∈ R [ x , . . . , x n ] and assume that V : = V ( f , . . . , f s ) ⊂ C n is equidi-mensional of dimension n − s. Suppose that g ∈ R [ x , . . . , x n ] satisfies the following conditions:1. Sing( V ) ∩ R n ⊂ V ( g ) ;2. dim ( V ∩ V ( g )) < n − s. hen the set of points where g restricted to V ∩ R n attains its extreme values intersects eachbounded connected component of (cid:0) V \ Sing( V ) (cid:1) ∩ R n . The proof of this theorem is based on the following lemma.
Lemma 3.2.
Let V be as in Theorem 3.1. Let g ∈ R [ x , . . . , x n ] such that dim ( V ∩ V ( g )) < n − s . Then, either ( V \ V ( g )) ∩ R n = ∅ or g restricted to V ∩ R n attains a non-zero extreme value oneach bounded connected component of ( V \ V ( g )) ∩ R n .Proof. Assume that ( V \ V ( g )) ∩ R n (cid:44) ∅ and let C be a bounded connected component of theset ( V \ V ( g )) ∩ R n . Since C (cid:49) V ( g ), there exists x ∈ C with g ( x ) (cid:44)
0. Let C be the Euclideanclosure of C so that C ⊂ V ∩ R n is closed and bounded, and g vanishes identically on C \ C .By the extreme value theorem, g attains both a minimum and a maximum on C . Since g is notidentically zero on C , either the minimum or the maximum value of g on C must be nonzero,so g attains a non-zero extreme value on C . Proof of Theorem 3.1.
Assume that (cid:0) V \ Sing( V ) (cid:1) ∩ R n (cid:44) ∅ . By Theorem 2.3, dim R V ∩ R n = n − s . By (2), ( V \ V ( g )) ∩ R n (cid:44) ∅ . By (1), ( V \ V ( g )) ∩ R n ⊂ (cid:0) V \ Sing( V ) (cid:1) ∩ R n , so the boundedconnected components of (cid:0) V \ Sing( V ) (cid:1) ∩ R n are subsets of the bounded connected componentsof ( V \ V ( g )) ∩ R n . By Lemma 3.2, g restricted to V ∩ R n attains a non-zero extreme value on eachbounded connected component of ( V \ V ( g )) ∩ R n yielding a point in every bounded connectedcomponent of (cid:0) V \ Sing( V ) (cid:1) ∩ R n .A lgorithm V ( f , . . . , f s ) is not equidi-mensional by using deformations and limits. However, the same algorithm can be used in theequidimensional case with input f , . . . , f s and a = ∈ Q s , i.e., without deformation. Example 3.3.
An example of a real curve with two singular cusps is often referred to as “Thom’slips,” e.g., f = y − ( x (1 − x )) as shown in Figure 1. An obvious choice of g which sat-isfies the conditions of Theorem 3.1 is g = x (1 − x ). Using Lagrange multipliers to opti-mize with respect to g results in two points (0 . , ± . g can be constructed algorithmically (see Section 4.2) yielding, e.g., g = x − x (1 − x )) + y which produces two points plotted as black circles, approximately(0 . , . . , − . (cid:0) V \ Sing( V ) (cid:1) ∩ R n . We note that the first choice of g demonstrates thatwhen Sing(V) is 0-dimensional, defining g as a product of a coordinate of these points will sat-isfy the conditions of Theorem 3.1. The second choice of g demonstrates the general methoddescribed in Section 4.2 which works in every dimension. Figure 1: “Thom’s lips” .2. Application to Kuramoto model The Kuramoto model from Kuramoto (1975) is a dynamical system used to model synchro-nization amongst n coupled oscillators. The maximum number of equilibria (i.e. real solutionsto steady-state equations) for n ≥ n = Theorem 3.4.
The maximum number of equilibria for the Kuramoto model with n = oscillatorsis . The steady-state equations for the n = f i ( θ ; ω ) = ω i − (cid:80) j = sin( θ i − θ j ) = , for i = , . . . , ω i ∈ R . Since only the angle di ff erences matter, onecan assume θ = = f + f + f + f = ω + ω + ω + ω , i.e., assume ω = − ( ω + ω + ω ). Substituting s i = sin( θ i ) and c i = cos( θ i ) yields F ( s , c ; ω ) = (cid:110) ω i − (cid:80) j = ( s i c j − s j c i ) , s i + c i − , for i = , , (cid:111) which is a polynomial system with variables s = ( s , s , s ) and c = ( c , c , c ), parameters ω = ( ω , ω , ω ), and constants s = c = F = ω varies over R . Let D ( ω ) be the discriminant polynomial of the system F , a polynomial in ω of degree 48. The number of real solutions of F is constant in each connected component of R \ V ( D ). Since it is easy to see that there can be no real solutions if | ω i | ≥ n − n = . R \ V ( D ). Applying Lemma 3.2 with f = g = D , i.e., by computing the real so-lutions of ∇ D = D (cid:44)
0, accomplishes this task. Exploiting symmetry and utilizing
Bertini (Bates et al.), alphaCertified (Hauenstein and Sottile (2012)), and
Macaulay2 (Grayson and Stillman) all solutions have been found and certified. In fact, this computationshowed that all real critical points of D arose, up to symmetry, along two slices shown in Fig-ure 2. A similar computation then counted the number of real solutions to F = dx.doi.org/10.7274/r0-5c1t-jw53 . We now consider the case when V ( f , . . . , f s ) is not equidimensional, i.e., it has some com-ponents of dimension greater than n − s . To handle this case, we perturb the polynomials byconstants and take limits. We present an algorithm that computes real smooth points on thislimit. This is applied to compute the real dimension of real algebraic sets in Section 3.4.We need the following from (Faug`ere et al., 2008, Lemma 1). Lemma 3.5.
Let f , . . . , f s ∈ R [ x , . . . , x n ] and fix l ≤ s and { i , . . . , i l } ⊂ { , . . . , s } . Then thereexists a Zariski closed subset A×E ⊂ C s × C such that for all ( a , . . . , a s ) ∈ R s \A and e ∈ R \E ,the ideal generated by the polynomials f i − ea i , . . . , f i l − ea i l is a radical equidimensional idealand V ( f i − ea i , . . . , f i l − ea i l ) is either empty or smooth of dimension n − l. Figure 2: Compact connected regions and critical points for the Kuramoto model with n = Definition 3.6.
Consider polynomials f , . . . , f s ∈ R [ x , . . . , x n ] and point a = ( a , . . . , a s ) ∈ Q s .We say that f , . . . f s and a satisfy Assumption (A) if (A) : There exists e > < e ≤ e , the polynomials f − ea , . . . , f s − ea s generate a radical equidimensional ideal and V a e : = V ( f − ea , . . . , f s − ea s ) is smooth andhas dimension n − s .We extend the results of Theorem 3.1 to the non-equidimensional case using deformationsand limits in A lgorithm lgorithm
1, the direction of the perturbation a ∈ Q s is part of the input because that is how we use it in the N umerical R eal D imension A lgorithm
2. However, by Lemma 3.5, for a generic a ∈ Q s , f , . . . , f s and a satisfies Assump-tion (A) . The subroutines of A lgorithm lgorithm 1 R eal S mooth P oint Input: n ≥ f , . . . , f s ∈ R [ x , . . . , x n ] and a ∈ Q s satisfying Assumption (A) . Let V a e : = V ( f − ea , . . . , f s − ea s ) and V : = lim e → + V a e . Output:
A finite set S ⊂ R n containing smooth points in each bounded connected componentof V ∩ R n that has dimension n − s . (1) Call the C omputation of g A lgorithm f , . . . , f s and a to obtain { ( g j , ( G j , L , W j )) : j = , ..., r } such that g j is in R [ x , . . . , x n ] and ( G j , L , W j ) is a deflatedwitness set for some V j ⊂ V that is a union of irreducible components of V .For each j = , . . . , r : (2) Set up the polynomial system L ( j ) e : = ∂ g j ∂ x i + s (cid:88) t = λ t ∂ f t ∂ x i : i = , . . . , n ∪ { f − ea , f − ea , . . . , f s − ea s } (2)in the variables x , . . . , x n , λ , . . . , λ s and parameter e . For the projection π x : C n × C s → C n ,compute the finite set U j : = lim e → π x ( V ( L ( j ) e )) ⊂ C n using the W itness P oints in L imits A lgorithm
3. Define T j : = U j \ V ( g j ) ∩ R n . (3) For each p ∈ T j , use the Membership Test of (Bates et al., 2013, Sec. 8.4) with input p and( G j , L , W j ) to find S j : = T j ∩ V j . (4) Return S : = (cid:83) rj = S j . Theorem 3.7.
Assume that f , . . . , f s ∈ R [ x , . . . , x n ] and point a ∈ Q s satisfies Assumption (A) .Then A lgorithm is correct. Furthermore, if S = ∅ , then V ∩ R n has no bounded connectedcomponents of dimension n − s. If S (cid:44) ∅ , then V ∩ R n has some connected components (possiblyunbounded) of dimension n − s.Proof. By Assumption (A) , V a e is smooth and equidimensional of dimension n − s for all suf-ficiently small e >
0. We can apply (Basu et al., 2006b, Prop. 12.38) to show that the set V = lim e → V a e ⊂ C n is a Zariski closed set that is either equidimensional of dimension n − s orempty. Assume that { ( g j , ( G j , L , W j )) : j = , ..., r } satisfies output specifications (i)-(vi) of A l - gorithm
5. Fix j ∈ { , . . . , r } and let V j ⊂ V be the union of irreducible components of V withwitness set ( G j , L , W j ). We note that since V a e is smooth and equidimensional for all su ffi cientlysmall e > V ( L je ) ⊂ C n × C n − d is zero dimensional, and so U j = lim e → π x ( V ( L ( j ) e )) is finite.Suppose ( V j \ V ( g j )) ∩ R n (cid:44) ∅ . Let C , . . . , C t ⊂ V j ∩ R n be the bounded connected componentsof V j ∩ R n where g j is not identically zero. Fix i ∈ { , . . . , t } . Since C i ⊂ V ∩ R n is compact,(Safey El Din and Tsigaridas, 2013, Prop. 5) shows that there exist connected components C ( e ) i , , . . . , C ( e ) i , s i of V a e ∩ R n for all su ffi ciently small e > C i = (cid:83) s i l = lim e → + C ( e ) i , l , each C ( e ) i , l is bounded, and ∪ s i l = C ( e ) i , l ∩ ∪ s j l = C ( e ) j , l = ∅ for all j (cid:44) i . For each l = , . . . , s i , let S ( e ) i , l : = π x ( V ( L ( j ) e )) ∩ C ( e ) i , l . By Lemma 3.2, S ( e ) i , l (cid:44) ∅ and it9ontains all points in C ( e ) i , l where g j takes its extreme values. Let S i : = (cid:83) s i l = lim e → S ( e ) i , l . Since S ( e ) i , l is bounded for all su ffi ciently small e , none of the limit points escape to infinity. Suppose that forall z ∈ S i we have g j ( z ) =
0. Since g j is not identically zero on C i , there exists z ∗ ∈ C i such that | g j ( z ∗ ) | >
0. Let z ∗ e ∈ C ( e ) i , l for some l = , . . . , s i such that lim e → z ∗ e = z ∗ . Then for any z ∈ S i ,if z e ∈ S ( e ) i such that lim e → z e = z , then for su ffi ciently small e we have that | g j ( z ∗ e ) | > | g j ( z e ) | .Since S i is finite, we can choose a common e value for all z ∈ S i so that if 0 < e < e then | g ( z ∗ e ) | > | g ( z e ) | for all z e ∈ S ( e ) i . Thus, S ( e ) i could not contain all points of C ( e ) i , l for l = , . . . , s i where g j takes its extreme values, a contradiction. So this proves lim e → π x ( V ( L ( j ) e )) ∩ C i = U j ∩ C i contains a point z ∈ C i such that g j ( z ) (cid:44) S j = U j \ V ( g j ) ∩ R n ∩ V j and S = (cid:83) rj = S j as in Steps (3) and (4). Since S j contains points in V j ∩ R n where g j is not zero, by (iii)-(vi) in A lgorithm V j ∩ R n , and also smooth in V ∩ R n . Thus if S (cid:44) ∅ , by Theorem 2.3, V ∩ R n musthave dimension n − s connected components. Conversely, if V ∩ R n has a bounded connectedcomponent of dimension n − s , then there exists j ∈ { , . . . , r } such that V j ∩ R n has a boundedconnected component of dimension n − s . By Theorem 2.3, this component has real smoothpoints. In fact, these real smooth points form a semi-algebraic set that has also dimension n − s .However, since dim (cid:16) V j ∩ V ( g j ) (cid:17) < n − s , g j does not vanish on all real smooth points of thiscomponent, but it vanishes on the singular points. By the above argument U j ∩ R n ∩ V j mustcontain points where g j is not zero, thus S j and S are not empty. Example 3.8.
Consider f , f ∈ R [ x , x , x ] where f = ( x + x + y + z −
1) and f = ( x + x + y + z − . Clearly, V ( f , f ) is not equidimensional, but V ( f , f ) ∩ R is compact of dimension 1. With a = (1 , lgorithm V is a curve with two irreducible components: V = V ( x + y + z − , x + y + z −
1) and V = V ( x + , x + y + z − x − y − z ). Using g = x − y and g = x (2 y − S = { (1 ± √ , ∓ √ , / } consisting of two smooth points on V ∩ R and S = ∅ . This section applies the R eal S mooth P oint A lgorithm polar varieties . The followinguses the notation in Safey El Din and Tsigaridas (2013). Definition 3.9.
Let f ∈ C [ x , . . . , x n ] be square-free and V = V ( f ) ⊂ C n . Consider the pro-jections π i ( x , . . . , x n ) = ( x , . . . , x i ) for i = , . . . , n . The polar variety associated to π i of V isdefined as crit( V , π i ) : = V (cid:32) f , ∂ f ∂ x i + , . . . , ∂ f ∂ x n (cid:33) ⊂ C n i = , . . . , n . emark 3.10. There is extensive literature about di ff erent notions of polar varieties (e.g., seeBank et al. (2010) for a survey). Here we use the simplest version following Safey El Din andTsigaridas (2013) and we reduce to the hypersurface case by taking a sum of squares. In practice,other notions of polar varieties may work better. We chose this presentation for conciseness. Inparticular, using more general forms of polar varieties would involve reproving (Safey El Dinand Tsigaridas, 2013, Propositions 2 and 3) (see the proof of Theorem 3.12 below).We use the following notation. Definition 3.11.
Let f ∈ R [ x , . . . , x n ], V = V ( f ) ⊂ C n , and A ∈ GL n ( R ). Then, we denote f A ( x ) : = f ( A x ), i.e. V ( f A ) is the image of V via the map x (cid:55)→ A − x .Our real dimension algorithm is as follows. Algorithm 2 N umerical R eal D imension Input: f , . . . , f s ∈ R [ x , . . . , x n ] such that V ( f , . . . , f s ) ∩ R n is compact where n ≥ Output:
The real dimension of V ( f , . . . , f s ) ∩ R n . (1) Choose a generic A ∈ GL n ( R ) and define f ( x ) : = s (cid:88) i = f Ai ( x ) ∈ R [ x , . . . , x n ] . Assume that for i = , . . . , n , (cid:16) f , ∂ f ∂ x i + , . . . , ∂ f ∂ x n (cid:17) and a : = e satisfy Assumption (A) . Let i : = n . (2) Using the R eal S mooth P oint A lgorithm (cid:16) f , ∂ f ∂ x i + , . . . , ∂ f ∂ x n (cid:17) and a : = e ,compute S ⊂ R n that contains smooth points in V ∩ R n , where V : = lim e → crit ( V ( f − e ) , π i ) . (3) If S (cid:44) ∅ then return i − (4) Set i : = i −
1. If i = −
1. If i > Theorem 3.12.
Let n ≥ , f , . . . , f s ∈ R [ x , . . . , x n ] such that V ( f , . . . , f s ) ∩ R n is compact.Then, A lgorithm is correct.Proof. Note that dim( V ( f , . . . , f s ) ∩ R n ) = dim( V ( f ) ∩ R n ). We cannot test if Assumption (A) issatisfied in Step (1). However, for a generic choice of A ∈ GL n ( R ) in Step (1), (cid:16) f , ∂ f ∂ x i + , . . . , ∂ f ∂ x n (cid:17) and a : = e satisfy Assumption (A) with probability one by (Safey El Din and Tsigaridas,2013, Prop. 2), so the input specification of A lgorithm n − i < n that we have the following loop invariant in Step (2):dim( V ( f ) ∩ R n ) ≤ i −
1. This is true when n − i =
0. Assume it is true for n − i < n , and we arein Step (2) with i >
0. By (Safey El Din and Tsigaridas, 2013, Prop. 3), V ∩ R n = V ( f ) ∩ R n for V : = lim e → crit ( V ( f − e ) , π i ) since dim( V ( f ) ∩ R n ) ≤ i − S (cid:44) ∅ , V has a real smooth point by Theorem 3.7, so by Theorem 2.3 we havedim( V ∩ R n ) = dim V = i − S = ∅ , the compactness of V ∩ R n andTheorem 3.7 implies that there are no real smooth points on V , so dim( V ∩ R n ) < dim V = i − i − = − V ( f ) ∩ R n = ∅ ,or we return to Step (2) with i − > .5. Benchmark Family for Real Dimension A benchmark family from Bannwarth and Safey El Din (2015) are hypersurfaces V ( f n ) ⊂ C n for n ≥ f n ( x , . . . , x n ) = (cid:16)(cid:80) nj = x j (cid:17) − (cid:80) nj = (cid:16) x j x j + (cid:17) (3)where x n + = x . Since f n is homogeneous, one knows dim V ( f n ) ∩ R n = dim( V ( f n , s n ) ∩ R n ) + s n = (cid:80) nj = x j − V ( f n , s n ) ∩ R n is compact. The cases 3 ≤ n ≤ ≤ n ≤
8. All code usedin these computations is available at dx.doi.org/10.7274/r0-5c1t-jw53 with the timingsreported using
Bertini (Bates et al.) on an AMD Opteron 6378 2.4 GHz processor using one(serial) or 64 (parallel) cores.For n = g = ∂ f /∂ x , one obtains smooth points on V ( f ) ∩ R thereby showingdim V ( f ) ∩ R = n = V ( f ) has multiplicity 2 with respect to f since f ( x , x , x , x ) = (cid:16) x − x + x − x (cid:17) . Trivially, a deflated witness system for V ( f ) is G = x − x + x − x . For g = x x , one obtainssmooth points on V ( f ) ∩ R showing dim V ( f ) ∩ R = n = , . . . ,
8, with g = ∂ f n /∂ x , one does not obtain smooth points on V ( f n ) ∩ R n showingdim V ( f n ) ∩ R n < n −
1. Therefore, one can move down the dimensions searching for real smoothpoints using perturbed polar varieties, similarly to Step (2) of Algorithm 2. Nonsingular realpoints are first found at dimension 2, i.e., dim V ( f n ) ∩ R n =
2. In fact, at dimension 2, thepolar variety contains various irreducible components of degree 2 and testing one is enough toconfirm the existence of a smooth real point. Table 1 lists the total computation time usingparallel processing. n dim V ( f n ) ∩ R n Time (min)5 2 3 .
636 2 5 .
737 2 34 .
818 2 159 . Table 1: Summary of benchmark problem (3) for 5 ≤ n ≤
4. Subroutines
This section describes the subroutines used in R eal S mooth P oint A lgorithm This subsection collects the numerical algebraic geometric tools needed for A lgorithm
Definition 4.1. If V ⊂ C n is equidimensional with dim V = k , a witness set for V is the triple( F , L , W ) such that 12 F ⊂ R [ x , . . . , x n ] is a witness system for V in that each irreducible component of V is anirreducible component of V ( F ), • L ⊂ R [ x , . . . , x n ] is a linear system where V ( L ) is a linear space of codimension k thatintersects V transversely, • W ⊂ C n is a witness point set which is equal to V ∩ V ( L ).If, in addition, each irreducible component of V has multiplicity one with respect to F , then F iscalled a deflated witness system and ( F , L , W ) is a deflated witness set .The first computation in our algorithms is to compute witness point sets of the limit V = lim e → + V ( f − a e , . . . , f s − a s e ) where f , . . . , f s and a = ( a , . . . , a s ) satisfy Assumption (A) .The di ffi culty is that V ( f − a e , . . . , f s − a s e ) is only smooth and equidimensional for 0 < e ≤ e where e is unknown and can be arbitrarily small. Instead, the next result shows that we canreplace e with t ξ where t ∈ (0 ,
1] and ξ ∈ C generic with | ξ | = ε . Let K = R or C and denote by K (cid:104) ε (cid:105) the field of Puiseux series over K , i.e. K (cid:104) ε (cid:105) : = (cid:88) i ≥ i a i ε i / q : i ∈ Z , q ∈ Z > , a i ∈ K . A Puiseux series z = (cid:80) i ≥ i a i ε i / q ∈ K (cid:104) ε (cid:105) is called bounded if i ≥ Proposition 4.2.
Let f , f , . . . , f s ∈ R [ x , . . . , x n ] , a = ( a , . . . , a s ) ∈ Q s and let ε be infinites-imal. Assume that V a ε : = V ( f − ε a , . . . , f s − ε a s ) ⊂ C (cid:104) ε (cid:105) n is smooth and equidimensionalof dimension n − s. Then for all but finiteley many ξ ∈ C with | ξ | = and for all t ∈ (0 , V a t ξ : = ( f − t ξ a , . . . , f s − t ξ a s ) ⊂ C n is smooth and equidimensional of dimension n − s and inthat case we have lim ε → V a ε = lim t → V a t ξ . Proof.
First, we show that for all but a finite number of choices of ξ ∈ C , V a ξ = V ( f − ξ a , . . . , f s − ξ a s ) is smooth. Note that from our assumptions on V a ε we get that f , . . . , f s and a satisfiesAssumption (A) for some e >
0. Consider the ideal using new variables x , z and λ , . . . , λ s : I : = (cid:104) f ( h )1 − a zx deg( f )0 , . . . , f ( h ) s − a s zx deg( f s )0 (cid:105) + (cid:104) ( λ ∇ ( f ) + . . . + λ s ∇ ( f s )) ( h ) (cid:105) . Here g ( h ) denotes the homogenization of g ∈ R [ x , . . . , x n ] by the variable x and ∇ is the di ff er-ential operator in the variables x , . . . , x n . Thus I is bi-homogeneous in the variables ( λ , . . . λ s )and ( x , . . . , x n ). Then the projection of X ( I ) ⊂ P n × P s × C onto C is a Zariski closed subsetof C , and since e is not in the projection, the projection is not C , thus a finite set Z . Clearly, for ξ ∈ C \ Z and for all p ∈ V a ξ , the Jacobian of f − ξ a , . . . , f s − ξ a s at p has rank s , thus V a ξ issmooth and equidimensional of dimension n − s . This also implies that for all but finitely many ξ ∈ C with | ξ | = t ∈ (0 ,
1] we have that V a t ξ = V ( f − t ξ a , . . . , f s − t ξ a s ) is smoothand equidimensional.Fix ξ ∈ C \ Z with | ξ | = V a t ξ is smooth and equidimensional. To prove the second claim,let L , . . . , L n − s ∈ C [ x , . . . , x n ] be linear polynomials such that L = V ( L , . . . , L n − s ) is a generic13inear space of codimension n − s which intersects both lim ε → V a ε and lim t → V a t ξ transversely. By ourassumptions, both V a ε ∩ L and V a t ξ ∩ L are finite. One can show that it is su ffi cient to prove thatlim ε → (cid:18) V a ε ∩ L (cid:19) = lim t → (cid:18) V a t ξ ∩ L (cid:19) to achieve the desired result.Let H ⊂ R [ x , . . . , x n , ε ] be the system H : = H ( x , ε ) = (cid:2) f − ε a , . . . , f s − ε a s , L , . . . , L n − s (cid:3) . Let S ⊂ C (cid:104) ε (cid:105) n be the finite set of bounded solutions of H =
0, where bounded is as defined forPuiseux series above. Then for all x ( ε ) ∈ S , let lim ε → x ( ε ) = x ∈ C n . Furthermore, by thedefinition of H , lim ε → S = lim ε → (cid:18) V a ε ∩ L (cid:19) . Since ε > x ( ε ) has an interval of convergence (0 , ε x ) ⊂ R forsome ε x >
0. Choose ε > ε < min x ∈ S ε x . Then, for z ∈ C with | z | ≤ ε , x ( z ) ∈ C n for x ∈ S .We consider the branch points of x ( z ) for all x ∈ S . In particular, the critical points C associated to these branch points are all z ∈ C such that there exists an x ∈ C n where H ( x , z ) = JH ( x , z ) =
0, where JH is the Jacobian matrix of H with respect to x . Then, since | S | < ∞ , we know |C| < ∞ ,Now let z ∈ C . Then there exists some ξ z ∈ S such that for t ∈ R , the path ξ z t passesthrough z , so that x ( t ξ z ) ∈ C n has some branching point. Let Z = { ξ z : z ∈ C} ⊂ S , since |C| < ∞ , | Z | < ∞ . Then, for ξ ∈ S \ Z , x ( t ξ ) ∈ C n for t ∈ (0 ,
1] does not pass through branchingpoints. Since S \ Z is Zariski dense in S , the same holds for generic ξ ∈ S .So let ξ ∈ S be generic and H ξ ⊂ C n + be the homotopy defined by the system H ξ : = H ξ ( x , t ) = (cid:2) f − t ξ a , . . . , f s − t ξ a s , L . . . , L n − s (cid:3) . The limit points of the solutions of H ξ are lim t → (cid:18) V a t ξ ∩ L (cid:19) . Let T ⊂ C n be the roots of H ξ ( x , | T | = | V a ε ∩ L| < ∞ . Furthermore, by the above argument the homotopy paths for H ξ areexactly described by the points in V a ε ∩ L ⊂ C (cid:104) ε (cid:105) n by replacing ε with t ξ . Hence,lim ε → (cid:18) V a ε ∩ L (cid:19) = lim t → (cid:18) V a t ξ ∩ L (cid:19) . This immediately yields A lgorithm ffi culty is thatthe limit could lie inside some irreducible component of V ( f , . . . , f s ) of dimension higher than n − s . Another di ffi culty is that the limit points may be singular, arising from multiple pathsconverging to the same limit point. These are demonstrated in the following. Example 4.3.
For f = x x , f = x x − x , and a = (1 , / , ⊂ V ( f , f ) = V ( x ). 14 lgorithm 3 W itness P oints in L imits Input: f , . . . , f s ∈ R [ x , . . . , x n ] and a = ( a , . . . , a s ) ∈ Q s satisfying Assumption (A) and L = { L , . . . , L n − s } ⊂ R [ x , . . . , x n ] generic linear polynomials. Output: W = V ( L ) ∩ V where V : = lim e → + V ( f − a e , . . . , f s − a s e ). (1) Choose generic ξ ∈ C with | ξ | = H ξ ( x , t ) = (cid:2) f − t ξ a , . . . , f s − t ξ a s , L . . . , L n − s (cid:3) . (2) Follow the finitely many homotopy paths V ( H ξ ( x , t )) starting for t = W consisting of the finite limit points of V ( H ξ ( x , t )) as t → ffi culties. Definition 4.4.
Let F ⊂ C [ y , . . . , y m ] and q ∈ V ( F ) ⊂ C m . The isosingular deflation opera-tor D is defined via ( F , q ) : = D ( F , q )where F ⊂ C [ y , . . . , y m ] consists of F and all ( r + × ( r +
1) minors of the Jacobian matrix JF for F where r = rank JF ( q ). Thus, q ∈ V ( F ), meaning that we can iterate this operator to con-struct a sequence of systems F j ⊂ C [ y , . . . , y m ] with ( F j , q ) = D ( F j − , q ) = D j ( F , q ) for j ≥ F ⊂ C [ y , . . . , y m ] is the isosingular deflation of F at q if there exists a minimal j ≥ F , q ) = D j ( F , q ) and dim NullSpace( JF ( q )) = dim F ( q ), where dim F ( q ) isthe maximal dimension of the irreducible components of V ( F ) containing q (called the localdimension of q with respect to F ).A lgorithm Algorithm 4 D eflated W itness S ystem Input: f , . . . , f s ∈ R [ x , . . . , x n ] and a = ( a , . . . , a s ) ∈ Q s satisfying Assumption (A) and p ∈ V : = lim e → + V ( f − a e , . . . , f s − a s e ), a generic point on a unique irreducible component V p of V . Output:
A deflated witness system G ⊂ R [ x , . . . , x n ] for V p . (1) For F ( x , t ) : = ( f − a t , . . . , f s − a s t ) and q = ( p , F of F at q . (2) Set G ( x ) = F ( x ,
0) and apply (Hauenstein and Wampler, 2013, Alg. 6.3) to compute theisosingular deflation G of G at p . Theorem 4.5.
Let f , . . . , f s , a , and p as in the input of A lgorithm . Then G, computed by A lgorithm , satisfies the output specifications.Proof. Since V p is an irreducible component of V , there exists an irreducible component Z ⊂ V ( F ( x , t )) ⊂ C n + such that V p ×{ } is an irreducible component of Z ∩ V ( t ) which is an intersec-tion. Hence, one can apply the isosingular deflation approach applied to intersections in (Hauen-stein and Wampler, 2017, Thm. 6.2). Although (Hauenstein and Wampler, 2017, Thm. 6.2)15ould deflate H ( x , t , t (cid:48) ) : = ( F ( x , t ) , t (cid:48) ) at q (cid:48) : = ( p , , t (cid:48) contained in H easily shows that one obtains an equivalent deflation as deflating F ( x , t ) at q = ( p , F ( x , t ). Therefore, V p must be an irreducible component of V ( F ( x , G ( x ) : = F ( x ,
0) is a witness system for V p . Since G need not be a deflated witness systemfor V p , one deflates G at p to yield a deflated witness system G for V p . One key aspect of A lgorithm g that satisfies the conditions of Theorem 3.1,i.e., Sing( V ) ∩ R n ⊂ V ( g ) and dim( V ∩ V ( g )) < dim( V ). There exist symbolic methods to computesuch a g for an irreducible variety V . For example, (Safey El Din et al., 2018, Lemma 4.3)computes the defining equation w of a generic projection π ( V ) that is a hypersurface. Then, g can be taken to be one of the partial derivatives of w . This idea could be extended to the casewhen V is not equidimensional using infinitesimal deformations and limits (c.f., Safey El Dinand Tsigaridas (2018)). A lgorithm g ’s depending on the isosingulardeflation sequence of the irreducible components. Algorithm 5 C omputation of g Input: f , . . . , f s ∈ R [ x , . . . , x n ] and a = ( a , . . . , a s ) ∈ Q s satisfying Assumption (A) . Let V a e : = V ( f − a e , . . . , f s − a s e ) and V : = lim e → + V a e . Output: r ≥
1, and (cid:110)(cid:16) g j , ( G j , L , W j ) (cid:17) : j = , . . . , r (cid:111) such that for all i (cid:44) j ∈ { , . . . , r } (i) g j ∈ R [ x , . . . , x n ], G j , L ⊂ R [ x , . . . , x n ], and W j ⊂ V .(ii) ( G j , L , W j ) is a deflated witness set of some V j ⊂ V , where V j is a union of irreduciblecomponents of V ;(iii) V = (cid:83) rj = V j (iv) Sing( V j ) ⊆ V ( g j )(v) dim( V j ) ∩ V ( g j ) < n − s (vi) dim( V i ∩ V j ) < n − s and V i ∩ V j ⊆ V ( g j ) . (1) Let L ⊂ R [ x , . . . , x n ] be a generic system of n − s linear polynomials. Compute W = V ∩ V ( L ) using W itness P oints in L imits A lgorithm
3. Let j : = (2) Fix p ∈ W and define W j : = { p } . Update W : = W \ { p } . (3) Using the D eflated W itness S ystem A lgorithm f , . . . , f s , a and p , compute G j ⊂ R [ x , . . . , x n ]. (4) For p (cid:48) ∈ W , if G j ( p (cid:48) ) = JG j ( p (cid:48) ) = s , then update W j = W j ∪ { p (cid:48) } and W = W \ { p (cid:48) } . (5) Let g j ( x ) : = det( M ( x )) where M is a generic rational linear combination of all s × s submatrices of JG j ( x ) . (6) If W (cid:44) ∅ , increment j = j + r = j and return.16 heorem 4.6. Let f , . . . , f s , a , V a e , and V be as in the input specification of A lgorithm . Then, A lgorithm is correct. In the proof we need the following definitions following Hauenstein and Wampler (2013).
Definition 4.7.
Let F ⊂ C [ y , . . . , y m ] and q ∈ V ( F ) ⊂ C m . Let D be the isosingular deflationoperator defined in Definition 4.4. We define • The deflation sequence of F at q is { d k ( F , q ) } ∞ k = where d k ( F , q ) = dnull( F k , q ) : = dim NullSpace JF k ( q ) with JF k the Jacobian matrix of F k with ( F k , q ) = D k ( F , q ). • Let V ⊂ V ( F ) be a non-empty irreducible algebraic set. Then V is an isosingular set of F if there exists a sequence { c k } ∞ k = such that V is an irreducible component of { p ∈ V ( F ) : d k ( F , p ) = c k , k ∈ N } . • Let V ⊂ V ( F ) be a non-empty irreducible algebraic set. Then Iso F ( V ) is the uniqueisosingular set with respect to F containing V such that Iso F ( V ) and V have the samedeflation sequence with respect to F . • Let V be an isosingular set for F . The set of singular points of V with respect to F isSing F ( V ) = (cid:110) p ∈ V : { d k ( F , p ) } ∞ k = (cid:44) { d k ( F , V ) } ∞ k = (cid:111) . Here, d k ( F , V ) is meant for a generic point in V . Proof.
By our assumption on the genericity of L , each point p ∈ W is a generic point of a uniqueirreducible components V p of V containing p . Then, G j ⊂ R [ x , . . . , x n ] computed in Step (3)deflates all generic points of V p . Step (4) adds all other points from W which are deflated by G j .In particular, every other point on V p contained in W will be added to W j . Hence, ( G j , L , W j ) is adeflated witness set for a union of irreducible components of V , denoted by V j , proving (ii). Since (cid:83) j W j = W , we also get (cid:83) j V j = V , which proves (iii). If y ∈ Sing( V j ), then rank( JG j ( y )) < s so all s × s minors of JG j ( y ) vanish. Hence, g j ( y ) = det( M ( y )) = p (cid:48) ∈ W j , some s × s minor of JG j ( p (cid:48) ) does not vanish at p (cid:48) . Since g j is a genericchoice of combinations of all such minors, g j ( p (cid:48) ) (cid:44) p (cid:48) ∈ W j . By Assumption (A) , V = lim e → V a e is equidimensional of dimension n − s , so for all p (cid:48) ∈ W , dim V p (cid:48) = n − s . Since g j does not vanish identically on V p (cid:48) for any p (cid:48) ∈ W j , we get dim( V j ) ∩ V ( g j ) < n − s , proving(v). To prove the first claim in (vi), note that each V i is a union of ( n − s )-dimensional irreduciblecomponents of V and sample points from the irreducible components of V are uniquely assignedto one W j . Then for i (cid:44) j , V i and V j cannot share an irreducible component, so their intersectionis lower dimensional.To prove the second claim in (vi) we use (Hauenstein and Wampler, 2013, Theorem 5.9)as follows. Let y ∈ V i ∩ V j . Suppose that X is an irreducible component of V i and Y is anirreducible component of V j such that y ∈ X ∩ Y . Let ξ ∈ C be generic with | ξ | = t a complexvariable, and denote f a ξ = f a ξ ( x , t ) : = ( f − a t ξ, . . . , f s − a s t ξ ). Then, X × { } and Y × { } areirreducible varieties of C n + and both are subsets of V ( f a ξ ) ⊂ C n + . Therefore, each is containedin a unique isosingular set of f a ξ denoted by Iso f a ξ ( X × { } ) and Iso f a ξ ( Y × { } ), respectively. Let F i ( x , t ) and F j ( x , t ) be their corresponding deflated witness systems, respectively. If F i = F j ,then I so F j ( x , ( X ) (cid:44) I so F j ( x , ( Y ) (otherwise X = Y ) so y ∈ Sing F j ( x , ( Y ). Note that by the D eflated itness S ystem A lgorithm G j ( x ) is the deflation of F j ( x ,
0) at a generic point of V j . Thisimplies by (Hauenstein and Wampler, 2013, Theorem 5.9) that y ∈ Sing G j ( Y ) and g j ( y ) = F i (cid:44) F j , then ( y ,
0) is in the intersection of two di ff erent isosingular sets so ( y ,
0) hasa di ff erent deflation sequence than Y × { } , i.e., ( y , ∈ Sing f a ξ ( Y × { } ). By (Hauenstein andWampler, 2013, Theorem 5.9), we have that ( y , ∈ Sing F j ( Y × { } ). Denoting the Jacobian by J : = JF j ( x , t ), we have that rank J ( y ) < s with rank J ( y (cid:48) ) = s for all generic y (cid:48) ∈ Y . Consider J (cid:48) : = JF j ( x , J corresponding to ∂ t removed). If rank J (cid:48) ( y (cid:48) ) = s for generic y (cid:48) ∈ Y , then G j = F j ( x , y ∈ Sing G j ( Y ), and g j ( y ) =
0. If rank J (cid:48) ( y (cid:48) ) < s for generic y (cid:48) ∈ Y ,we claim that rank J (cid:48) ( y ) < rank J (cid:48) ( y (cid:48) ) for generic y (cid:48) ∈ Y . First note that both rank J f ( y ) ≤ s − J f ( y (cid:48) ) ≤ s − f = ( f , . . . , f s ), so without loss of generality, we assume that ∇ f ( y ) = ∇ f ( y (cid:48) ) =
0. Note that the ∂ t column of J = JF j ( x , t ) has the only possibly non-zero constant entries in the rows corresponding to f − a t ξ, . . . , f s − a s t ξ . Denote by J (cid:48)(cid:48) thesubmatrix of J (cid:48) with the row corresponding to f removed. Then for a generic y (cid:48) ∈ Y we haverank J (cid:48) ( y (cid:48) ) = s −
1, since among all s × s minors of J ( y (cid:48) ) some has to be non-zero, and the onlypossible non-zeros are the ones that are a times the ( s − × ( s −
1) minors of J (cid:48)(cid:48) ( y (cid:48) ), thus a (cid:44) J (cid:48) ( y (cid:48) ) = s −
1. On the other hand, the s × s minors of J ( y ) contain all ( s − × ( s −
1) minorsof J (cid:48)(cid:48) ( y ) times a , so all these minors of J (cid:48)(cid:48) ( y ) must be zero. This implies that rank J (cid:48) ( y ) < s − J (cid:48) ( y ) < rank J (cid:48) ( y (cid:48) ). In particular, y ∈ Sing F j ( x , ( Y ) and by (Hauenstein and Wampler,2013, Theorem 5.9), y ∈ Sing G j ( Y ) which implies that g j ( y ) = lgorithms f , . . . , f s . On the other hand, the degree of the polynomial w computed in the symbolic approach in (Safey El Din et al., 2018, Lemma 4.3) mentioned aboveis the degree of V bounded by the product of the degrees of the input polynomials. Nonetheless,the disadvantage of our approach is that in the worst case, we need as many iterations in thedeflation as the multiplicity of the points and this may result polynomials that are higher degreethan w . We have the following bound on the degree of g as a function on the number of iterationsin the deflation: Proposition 4.8.
Let f = ( f , . . . , f s ) and a = ( a , . . . , a s ) ∈ Q s such that V a e : = V ( f − a e , . . . , f s − a s e ) satisfies Assumption (A) . Let D : = max si = { deg( f i ) } and fix p ∈ V : = lim e → V a e .If A lgorithm takes k iterations of the isosingular deflation to output G ⊂ R [ x , . . . , x n ] , thedegrees of the polynomials in G are bounded by s k D. Furthermore, if g ( x ) : = det( M ( x )) ∈ R [ x , . . . , x n ] where M ( x ) is a s × s submatrix of JG ( x ) , then deg( g ) ≤ s k + D.Proof.
The first claim follows from the fact that each iteration of the deflation algorithm adds theminors of the Jacobian of the polynomials in the previous iteration, and these minors have sizeless than s . Thus, the degrees of polynomials added to the system in each iteration are at most s times the degrees of the polynomials in the previous iteration. The second claim follows fromthe first.Using Proposition 4.8, we can bound the number of homotopy paths followed in Step (2)in the R eal S mooth P oint A lgorithm
1, which is the bottleneck of our method. Note that thenumber of iterations r is at most deg( V ) ≤ D n and the Membership Test of (Bates et al., 2013,Sec. 8.4) utilized in Step (3) follows at most | W j | = deg( V j ) ≤ deg( V ) ≤ D n homotopy paths.18 orollary 4.9. Let f = ( f , . . . , f s ) and a be as above. Consider the zero-dimensional polynomialsystem L ( j ) e for some fixed j ∈ { , . . . , r } as in Step (2) of A lgorithm . Then, the number ofcomplex roots of L ( j ) e is bounded by deg( g j ) n D s ≤ s ( k j + n D n + s , where D is as above when weassume that deg( g j ) ≥ D and k j is the number of iterations of the isosingular deflation needed tocompute G j using A lgorithm . Acknowledgments
The authors thank Mohab Safey El Din and Elias Tsigaridas for many discussions regard-ing real algebraic geometry. This research was partly supported by NSF grants CCF-1812746(Hauenstein) and CCF-1813340 (Szanto and Harris).
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