On sign conditions over real multivariate polynomials
arXiv:0801.0586v1 [math.AG] 3 Jan 2008
On sign conditions over real multivariate polynomials Gabriela Jeronimo∗† Daniel Perrucci∗
Juan Sabia∗
Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina
Abstract We present a new probabilistic algorithm to find a finite set of points intersecting the closure of each connected component of the realization of every sign condition over a family of real polynomials defining regular hypersurfaces that intersect transversally. This enables us to show a probabilistic procedure to list all feasible sign conditions. We extend our main algorithm to the case of an arbitrary multivariate polynomial and to that of an arbitrary family of bivariate polynomials. The complexity bounds for these procedures improve the known ones.
Keywords: Real multivariate polynomials, sign conditions and consistency problem, algorithms and complexity.
1
Introduction
Given f1 , . . . , fm ∈ R[x1 , . . . , xn ], a sign condition σ ∈ {}m is said to be feasible if the system f1 σ1 0, . . . , fm σm 0 has a solution in Rn , and the set of solutions is called the realization of σ. One of the basic problems in ∗ Partially supported by the following Argentinian research grants: UBACyT X112 (2004-2007), UBACyT X847 (2006-2009), CONICET PIP 5852/05 † Partially supported by ANPCYT PICT 2005 17–33018.
1
computational semialgebraic geometry is to decide whether a sign condition is feasible. This problem is a particular case of quantifier elimination and, on the other hand, many elimination algorithms use subroutines deciding all the feasible sign conditions for a family of polynomials. The first elimination algorithms over the reals are due to Tarski [37] and Seidenberg [35] but their complexity is not elementary recursive. Collins [14] was the first to obtain a doubly exponential complexity. In [18], Grigor’ev and Vorobjov presented an algorithm with single exponential complexity to decide the consistency of systems of inequalities by studying the critical points of a function. This idea was also used to obtain more efficient quantifier elimination procedures ([21, 28, 7]). The algorithms rely on the computation of a finite set of points intersecting every connected component of a semialgebraic set. A standard technique was to take sums of squares and introduce infinitesimals to study only smooth and compact hypersurfaces. The specific problem of consistency for equalities over R was treated through the critical point method afterwards. In [30], the case of a single equation is studied, reducing the introduction of infinitesimals, and in [2], an algorithm with no infinitesimals is given to deal with arbitrary positive dimensional systems. Several probabilistic procedures lead to successive improvements. Using classical polar varieties, in [3, 4], the case of a smooth compact variety given by a regular sequence is tackled within a complexity depending polynomially on an intrinsic degree of the systems involved and the input length. To achieve this complexity, straight-line programs and an efficient procedure to solve polynomial equation systems over C ([15]) are used. The compactness assumption is dropped in [5, 6] by using generalized polar varieties. The non-compact case is also considered in [32] for a smooth equidimensional variety defined by a radical ideal by studying projections over polar varieties, and an extension to the non-equidimensional situation is given in [33]. Finally, [31] deals with sets of the type {f > 0} through the computation of generalized critical points. In this paper, we present a probabilistic algorithm that, given polynomials, obtains a parametric description of a finite set of points intersecting the closure of each connected component of the realization of any sign condition over them in three cases: firstly, when the polynomials define regular hypersurfaces in Cn intersecting transversally (see Hypothesis 4.1); secondly, for a single arbitrary multivariate polynomial, and finally, for a family of arbitrary bivariate polynomials. In the first case, the output of the algorithm allows to obtain all the feasible sign conditions by using the techniques in [13]. 2
The input and intermediate computations in our algorithms are encoded by straight-line programs (see Section 2.2). The output is described by means of geometric resolutions, that is to say, by univariate rational parametrizations of 0-dimensional varieties (see Section 2.3). The sketch of the algorithm is the following: given polynomials f1 , . . . , fm , a generic change of variables ensures that each connected component C ⊂ Rn of each feasible sign condition over them avoids some asymptotic behavior with respect to the projection on the first coordinate (see Section 3.1). More precisely, either the projection of C on x1 is R or the projection of C is a proper closed interval. In this last case, points in C are obtained as extremal points of x1 . These points are solutions of particular systems of equations (see Section 3.2) which are dealt with by means of deformation techniques (see Section 3.3) to obtain geometric resolutions of finite sets including them. When the projection is R, C is intersected with x1 = p1 for some value p1 and the algorithm follows recursively. The following theorem summarizes our main results: Theorem Let K be a subfield of R. There is a probabilistic algorithm that, from f1 , . . . , fm ∈ K[x1 , . . . , xn ] defining regular hypersurfaces in Cn which intersect transversally, with degrees bounded by d ≥ 2 and encoded by a straight-line program of length L, computes a finite family of geometric resolutions of 0-dimensional varieties whose union contains at least one point in the closure of each connected component of the realization of every sign conPmin{m,n} m n−1 n 2 dition over f1 , . . . , fm within O d (L + d) operations s=1 s s−1 in K up to logarithmic factors. The same result holds in the case of a single polynomial without any hypothesis and of an arbitrary family of bivariate polynomials. n The factor n−1 d in the complexity is just an upper bound for the s−1 bihomogeneous B´ezout numbers (see [36]) of the systems arising from the Lagrange characterization of critical points of a map (cf. [33]). One of the new tools to achieve our complexity order, which improves the previous ones depending on these parameters, is the use of algorithmic deformation techniques specially designed for bihomogeneous systems (see [19] for a similar approach). Up to now, these systems were handled with general algorithms for solving complex polynomial equations (see, for instance, [1, 29, 17, 26]). Our work can also be seen as an extension of [32] and [6] in the sense that we deal not only with equations but also with inequalities. In the three situations previously mentioned, our algorithm may be adapted 3
to deal with a particular sign condition within a lower complexity order. Moreover, for polynomials defining regular hypersurfaces which intersect transversally, we also give a procedure for computing the list of all feasible sign conditions over them from the output of our main algorithm with Pmin{m,n} m n−1 n Ω n−1 n O d m + d L additional operations (up to s=1 s s−1 s−1 logarithmic factors) where Ω > 2 is a number such that two ℓ × ℓ matrices can be multiplied with O(ℓΩ ) operations. For a treatment of the same problem in full generality, see for instance [8]. As in the case of bivariate polynomials, we expect our deformation approach will work in the general multivariate setting without any assumptions. This is subject of our current research. This paper is organized as follows: In Section 2 we state some preliminary notions. Section 3 is devoted to presenting the basic tools used in the design of our algorithms: first, we show some properties of generic changes of variables; then, we present the Lagrange-type polynomial equation systems characterizing critical points, and finally, the algorithmic deformation techniques for bihomogeneous systems are shown. In Section 4, we present our main algorithms.
2 2.1
Preliminaries Notation
Throughout this paper Q, R and C denote the fields of rational, real and complex numbers respectively and K denotes an effective subfield of R. If k is a field, k¯ denotes an algebraic closure of k. If A is a set in a topological space, A denotes its closure. For a positive integer n and an algebraically closed field k, we denote by Ank and Pnk (or by An or Pn if k is clear from the context) the n-dimensional affine and projective space over k respectively, equipped with their Zariski topologies. We adopt the usual notion of dimension and degree for algebraic varieties (see for instance [36]). Let Π1 : Rn → R be the projection on the first coordinate. For any nonempty set A ⊂ Rn , we define Zi (A) = {(x1 , . . . , xn ) ∈ A | x1 = inf Π1 (A)} if Π1 (A) is bounded from below, and Zi(A) = ∅ otherwise. Similarly, Zs (A) = {(x1 , . . . , xn ) ∈ A | x1 = sup Π1 (A)} if Π1 (A) is bounded from above, and Zs (A) = ∅ otherwise. Finally, we denote Z(A) = Zi (A) ∪ Zs (A). Given f1 , . . . , fm ∈ R[x1 , . . . , xn ], C will denote the set of all the connected 4
components of the realizations of all feasible sign conditions σ ∈ {}m over f1 , . . . , fm .
2.2
Algorithms and complexity
The algorithms we consider are described by arithmetic networks over K (see [39]). The complexity of an algorithm is the number of operations and comparisons in K. Our algorithms are probabilistic in the sense that they depend on some random choices of points and their output is correct whenever these points lie outside proper Zariski closed sets. The objects we deal with are polynomials with coefficients in K. Each polynomial is represented either as the array of all its coefficients in a prefixed order of all the monomials, or by a straight-line program. Roughly speaking, a straight-line program (or slp, for short) over K encoding a list of polynomials in K[x1 , . . . , xn ] is an arithmetic circuit which enables us to evaluate these polynomials at any given point in Kn . The number of instructions is called the length of the slp (for a precise definition we refer to [12, Definition 4.2]).
2.3
Geometric resolutions
A way of representing 0-dimensional affine varieties that is widely used in computer algebra nowadays is a geometric resolution. This notion, which was first introduced by Kronecker [25] and K¨onig [24], appears in the literature under different names such as rational univariate representation, shape lemma, etc. (for a historical account, see [17]). The precise definition we use is the following: Let k be a field and V = {ξ (1) , . . . , ξ (D) } a 0-dimensional variety in Ank¯ defined by polynomials in k[x1 , . . . , xn ]. Given a separating linear form ℓ ∈ k[x1 , . . . , xn ] for V (that is, ℓ(ξ (i) ) 6= ℓ(ξ (j) ) if i 6= j), V is characterized by: the Qif U is a new variable, (i) minimal polynomial p := 1≤i≤D (U − ℓ(ξ )) ∈ k[U] of ℓ over V , a polynomial p˜ ∈ k[U] with deg(˜ p) < D and relatively prime to p, and polynomials w1 , . . . , wn ∈ k[U] with deg(wj ) < D for every 1 ≤ j ≤ n satisfying n w o wn ¯ n 1 ¯ p(η) = 0 . V = (η), . . . , (η) ∈ k | η ∈ k, p˜ p˜
The family {p, p˜, w1 , . . . , wn } ⊂ k[U] is called a geometric resolution of V (associated with ℓ). As p˜ is invertible in k[U]/(p(U)), this is equivalent 5
to the standard notion a family {p, v1 , . . . , vn } in of geometric resolution: n ¯ p(η) = 0 . We will k[U] satisfying V = v1 (η), . . . , vn (η) ∈ k | η ∈ k, use both definitions alternatively, since the complexity of passing from one representation to the other does not modify the overall complexity of our algorithms.
3
General approach
3.1
Avoiding asymptotic situations
Notation 3.1 For 1 ≤ k ≤ n, Πk : Rn → R is the projection Πk (x1 , . . . , xn ) = xk . For A ⊂ Rn , Zi (A, k) = {(x1 , . . . , xn ) ∈ A | xk = inf Πk (A)} if A is bounded from below and Zi (A, k) = ∅ otherwise. Similarly, Zs (A, k) = {(x1 , . . . , xn ) ∈ A | xk = sup Πk (A)} if A is bounded from above and Zi (A, k) = ∅ otherwise. Finally, Z(A, k) = Zi (A, k)∪Zs (A, k). In particular, Zi (A, 1) = Zi (A), Zs (A, 1) = Zi (A) and Z(A, 1) = Z(A) (see Section 2.1). Given f1 , . . . , fm ∈ R[x1 , . . . , xn ] and p ∈ Rn , C(k, p) is the set of all the connected components of the Rn -subsets C ∩ {x1 = p1 , . . . , xk−1 = pk−1 } with C ∈ C. In particular, C(1, p) = C for every p ∈ Rn (see Section 2.1). The non-asymptotic behavior for every step of the algorithm ensured by a generic change of variables is stated in: Proposition 3.2 Let f1 , . . . , fm ∈ R[x1 , . . . , xn ]. Then, after a generic change of variables over Q, for every p ∈ Rn , 1 ≤ k ≤ n and C ∈ C(k, p), Z(C, k) is finite (possibly empty), and if Πk (C) is bounded from below or above, then Z(C, k) 6= ∅. To prove Proposition 3.2, we will use the following: Lemma 3.3 Let {fij }1≤i≤n,1≤j≤li ⊂ R[x1 , . . . , xn ] be a family of nonzero polynomials satisfying simultaneously: a) for 1 ≤ i ≤ n, {fij }1≤j≤li ⊂ R[x1 , . . . , xi ] is closed under derivation with respect to xi and every fij is quasi-monic (that is, monic up to a constant) in xi , b) for 1 ≤ i ≤ n, there exists 1 ≤ li′ ≤ li such that for 1 < i ≤ n, ′ {f(i−1)j }1≤j≤li−1 slices {fij }1≤j≤li in the sense of [11, Definition 2.3.4]. 6
Let p ∈ Rn and let 1 ≤ k ≤ i ≤ n. Let D ⊂ Ri be defined by a boolean formula on the polynomials fij , 1 ≤ j ≤ li′ , involving equalities and inequalities to zero, and let C be a connected component of D ∩ {x1 = p1 , . . . , xk−1 = pk−1 }. Then the set Z(C, k) is finite (possibly empty). Moreover, if Πk (C) is bounded from below or above, then Z(C, k) 6= ∅. Proof. As {fij (p1 , . . . , pk−1 , xk , . . . , xn )}k≤i≤n,1≤j≤li in R[xk , . . . , xn ] is a family that satisfies the hypotheses (1 ≤ k ≤ n), it is enough to prove the lemma for k = 1. For i = 1, the result is clear. Suppose the statement is true for i − 1. Let Π : Ri → Ri−1 be the projection on the first i − 1 coordinates. Following the notation in [11, Ch. 2], let A1 , . . . , Aℓ be the slicing of Ri−1 with respect to fi1 , . . . , fili ′ and, for 1 ≤ s ≤ ℓ, let ξs,1 < · · · < ξs,as : given by f(i−1)1 , . . . , f(i−1)li−1 As → R be the continuous semialgebraic functions that slice As × R. Let ′ , . . . , f(i−1)li−1 As,1 , . . . , As,us be the connected S components of As . As f(i−1)1S slice fi1 , . . . , fili′ , Π(C) = h Ash ,uh , and then Z(Π(C)) ⊂ h Z(Ash ,uh ). Since each Ash can be described by a boolean formula only involving the ′ , by inductive hypothesis, Z(Π(C)) is finite. polynomials f(i−1)1 , . . . , f(i−1)li−1 Now, if w ∈ Z(C), Π(w) ∈ Z(Π(C)). Moreover, fi1 , . . . , fili′ are quasi-monic in xi and at least one of them vanishes at w. Therefore, Z(C) is a finite set. Suppose now that Π1 (C) is an interval bounded, for example, from below. Then, there exists z = (z1 , . . . , zi−1 ) ∈ Zi (Π(C)) ⊂ Π(C). Let γ : [0, 1] → Ri−1 be a continuous semialgebraic curve such that γ((0, 1]) ⊂ Π(C) and γ(0) = z (see [11, Theorem 2.5.5]). We may suppose that for 0 < ε ≤ 1, γ((0, ε]) ⊂ A1 . Using [11, Lema 2.5.6], each ξ1,a can be extended to a semialgebraic continuous function ξ 1,a over A1 . Let x˜ := γ(ε) and y ∈ R such that (˜ x, y) ∈ C. Depending on the position of y with respect to ξ 1,1 (˜ x) < x), it is easy in any case to define a continuous semialgebraic · · · < ξ 1,a1 (˜ function h : [0, ε] → R such that γ˜ : [0, ε] → Ri defined as γ˜ (t) = (γ(t), h(t)) satisfies γ˜ ((0, ε]) ⊂ C (note that the signs of fi1 , . . . , fili are constant over γ˜ ((0, ε])) and, therefore, (z, h(0)) = γ˜ (0) ∈ C. As z1 = inf Π1 (Π(C)) = inf Π1 (C), (z, h(0)) ∈ Zi (C). Now, we can prove Proposition 3.2: Proof. Let V := (vr,s )1≤r,s≤n be a matrix whose entries are new variables and consider the family of polynomials {Fij }1≤i≤n,1≤j≤li ⊂ R[v, x] defined in the following way: 7
• Take ln′ = m and, for 1 ≤ j ≤ ln′ , let Fnj (V, x) = fj (V x). Then, for 1 ≤ j0 ≤ ln′ , if degx Fnj0 = dnj0 , add its first dnj0 − 1 derivatives with respect to xn to the list to obtain the family {Fnj }1≤j≤ln . • From {F(i0 +1)j }1≤j≤li0 +1 ⊂ R[v, x1 , . . . , xi0 +1 ], first, take all possible resultants and subresultants with respect to xi0 +1 between pairs of elements to form {Fi0 j }1≤j≤li′ ⊂ R[v, x1 , . . . , xi0 ], not considering the ones that are iden0 tically zero. Then, for 1 ≤ j0 ≤ li′0 , if degx Fi0 j0 = di0 j0 , add its first di0 j0 − 1 derivatives with respect to the variable xi0 to obtain the family {Fi0 j }1≤j≤li0 . From the recursive definition of {Fij }1≤i≤n,1≤j≤li , it can be shown induc A 0 tively that for 1 ≤ j ≤ li′ and A ∈ Qi×i , Fij (V, Ax) = Fij V ,x , 0 In−i B0 B 0 ′ (i−1)×(i−1) and for li +1 ≤ j ≤ li and B ∈ Q , Fij V, x = Fij V ,x . 0 1
0 In−i+1 d
Using these identities, if dij := degx Fij and qij ∈ R[v] is the coefficient of xi ij in Fij (1 ≤ i ≤ n, 1 ≤ j ≤ li ), the existence of a linear change of variables making Fij ′ (1 ≤ j ′ ≤ li′ ) quasi-monic in xi implies that qij 6≡ 0. Let U = {V0 ∈ Cn×n | qij (V0 ) 6= 0 for 1 ≤ i ≤ n, 1 ≤ j ≤ li }. By [9, Proposition 4.34 and Theorem 5.14], for every V0 ∈ Qn×n ∩ U, the set {fij (x)}1≤i≤n,1≤j≤li defined by fij (x) = Fij (V0 , x) satisfies both conditions in Lemma 3.3 and fnj = fj (V0 x) for 1 ≤ j ≤ ln′ . The result follows.
The following proposition is a major tool for our algorithm (cf. [32, Theorem 2]). n Proposition , after a generic change of variables, the Sn 3.4S For every p ∈ R set {p} ∪ k=1 C∈C(k,p) Z(C, k) is finite and has at least one point in the closure of each connected component of the realization of each feasible sign condition over the polynomials f1 , . . . , fm .
Proof. By Proposition 3.2 the set is finite. Let C ∈ C. Then, either Z(C) is a finite non-empty set and intersects C or Π1 (C) = R (see again Proposition 3.2). Now, it suffices to consider the connected components of C ∩ {x1 = p1 } and the projection Π2 . The result follows recursively. This is the recursion in our algorithm. Note that, for 2 ≤ k ≤ n, the k-th variable can be seen as the first one for fj (p1 , . . . , pk−1, xk , . . . , xn ) (1 ≤ j ≤ m). Therefore, S it suffices to consider the problem of finding a finite set which contains C∈C Z(C). 8
3.2
Equations defining extremal points
Let f1 , . . . , fm ∈ R[x1 , . . . , xn ] and let S := {i1 , . . . , is } ⊂ {1, . . . , m}. If 1 ≤ s < n, the implicit function theorem implies that the points with maximum or minimum first coordinate x1 in a connected component of {fi1 = · · · = fis = 0} satisfy ( fi1 (x) = · · · = fis (x) = 0, Ps (1) n−1 j=1 µj ∇fij (x) = (0, . . . , 0) ∈ R
for µ1 , . . . , µs ∈ R not simultaneously zero, where ∇fij (x) denotes the vector obtained by removing the first coordinate from the gradient vector ∇fij (x). System (1) can be considered for any s ≤ m and any {i1 , . . . , is } ⊂ {1, . . . , m} and defines a variety WS ⊂ AnC ×PCs−1 . Let Π denote the projection Π : AnC × PCs−1 → AnC .
Remark 3.5 If s ≥ n or s = 1, Π(WS ) is the solution set of fi1 (x) = · · · = ∂f ∂f fis (x) = 0 or fi1 (x) = ∂xi21 (x) = · · · = ∂xin1 (x) = 0 respectively. We will characterize Π(WS ) by means of these simpler systems in these cases. The following result is an adaptation of the Karush-Kuhn-Tucker conditions for non-linear optimization (see [27, Chapter 3]) which generalize the Lagrange multipliers theorem: Proposition 3.6 If C is a connected componentSof the set {f1 = · · · = fq = 0, fq+1 > 0, . . . , fm > 0} ⊂ Rn , then Z(C) ⊂ Π {1,...,q}⊂S⊂{1,...,m}, WS . S6=∅
Proof. Let z = (z1 , . . . , zn ) ∈ Zi (C). Note that {i ∈ {1, . . . , m} | fi (z) = 0} is not empty and, without loss of generality, assume it equals {1, . . . , t} with q ≤ t ≤ m. Suppose that {∇fi (z), 1 ≤ i ≤ t} is linearly independent. Let f = (f1 , . . . , ft ) : Rn → Rt . We may suppose that the minor corresponding to the variables n − t + 1, . . . , n in the Jacobian matrix Df (z) is not zero. Applying the inverse function theorem to h(x) = (x1 − z1 , . . . , xn−t − zn−t , f1 (x), . . . , ft (x)), there exist an open neighborhood U of z, ε ∈ R>0 and a map g : (−ε, ε)n → U inverse to h : U → (−ε, ε)n . Let w ∈ C ∩ U and y = (y1 , . . . , yn ) = h(w). Let γ : [−ε/2, y1 ] → (−ε, ε)n be γ(u) = (u, y2, . . . , yn ). Then, Im(g ◦ γ) is a connected curve and g ◦ γ(y1 ) = w ∈ C. On the other hand, for 1 ≤ i ≤ q, fi (g ◦ γ(u)) = 0 and for q + 1 ≤ i ≤ t, fi (g ◦ γ(u)) > 0. This implies that Im(g ◦ γ) ⊂ {f1 = 9
· · · = fq = 0, fq+1 > 0, . . . , fm > 0} (for t + 1 ≤ i ≤ m, fi (z) > 0 and therefore we may suppose fi > 0 over U). Thus, Im(g ◦ γ) ⊂ C, but the first coordinate of g ◦ γ(−ε/2) is −ε/2 + z1 < z1 . This contradicts the fact that z1 = inf Π1 (C). Therefore, {∇fi (z), 1 ≤ i ≤ t} should be linearly dependent and z ∈ Π(W{1,...,t} ).
3.3
Deformation techniques for bihomogeneous systems
In this section we describe a procedure based on the symbolic deformation techniques developed in [16, 15, 17, 20, 34], adapted to the bihomogeneous setting following [19], that we will apply to deal with the specific polynomial systems of type (1). 3.3.1
The deformation
We consider a system of equations given by polynomials in K[x1 , . . . , xn , µ1 , . . . , µs ]: f1 (x) = 0, . . . , fs (x) = 0, fs+1 (x, µ) = 0, . . . , fr (x, µ) = 0 where 2 ≤ s ≤ n − 1 and r = s + n − 1, such that, for 1 ≤ i ≤ r, degx fi := di ≤ d and, for s+1 ≤ i ≤ r, fi is homogeneous of degree 1 in the variables µ. (To avoid introducing notation, we call these polynomials f1 , . . . , fr ; however, they will not be the input polynomials of our algorithms but new polynomials obtained as in (1)). Let W ⊂ AnC × PCs−1 be the variety they define. Let g1 (x), . . . , gs (x), gs+1(x, µ), . . . , gr (x, µ) ∈ K[x, µ] with the same degrees and homogeneity as f1 , . . . , fr such that they define a 0-dimensional variety in An × Ps−1 and their homogenizations in x with a new variable x0 define a variety in Pn × Ps−1 which is contained in {x0 6= 0}. For every 1 ≤ i ≤ r, let Fi := gi + (t − 1)(gi − fi ). Consider the variety Vˆ ⊂ A×An ×Ps−1 defined by F1 , . . . , Fr , and write Vˆ = V (0) ∪V (1) ∪V, where V (0) is the union of the irreducible components of Vˆ contained in {t = 0}, V (1) is the union of those contained in {t = t0 } for some t0 ∈ C \ {0}, and V is the union of the remaining ones. In Section 4, given a family of m polynomials in K[x], for S ⊂ {1, . . . , m}, we will add a subscript S in the notation Vˆ , V (0) , V (1) and V to indicate that these varieties are defined from the bihomogeneous system (1) associated with S. Lemma 3.7 Under the previous hypotheses, Vˆ ∩ {t = 1} = V ∩ {t = 1} and every irreducible component of V is 1-dimensional and intersects {t = 1}. 10
Proof. Each irreducible component of Vˆ has dimension at least 1 and Vˆ ∩ {t = 1} is 0-dimensional. Therefore, Vˆ ∩ {t = 1} = V ∩ {t = 1}. Let V1 be an irreducible component of V and V1 its Zariski closure in 1 A × Pn × Ps−1 . The projection of V1 to A1 is onto and so, V1 ∩ {t = 1} = 6 ∅. But our assumption on g1 , . . . , gr implies that V1 ∩ {t = 1} = V1 ∩ {t = 1}. Then V1 ∩ {t = 1} = 6 ∅ and it is 0-dimensional. It follows that dim(V1 ) = 1. The procedure we are going to apply will enable us to compute a geometric resolution of a finite subset of the variety Π(W ) defined by our input system. Let π : A × An × Ps−1 → An × Ps−1 be the projection (t, x, µ) 7→ (x, µ). Corollary 3.8 With our previous assumptions, π(V ∩ {t = 0}) is a finite subset of W containing its isolated points. In what follows, we introduce some polynomials g1 , . . . , gr we will deal with and show some of their basic properties. Qi (xi − j). For s + 1 ≤ i ≤ r, for Notation 3.9 For 1 ≤ i ≤ s, gi (x) := dj=1 Pn 1 1 1 ≤ j ≤ di , if φ(i,j)(x) := k=s+1 (i−s−1)d+j−1+k−s xk + (i−s−1)d+j−1+n+1−s , Ps Qdi 1 then gi (x, µ) := k=1 i−s−1+k µk . j=1 φ(i,j) (x)
Lemma 3.10 Let g1 , . . . , gr be as in Notation 3.9. Then: (i) The system Q g1 = · ·· =Pgr = 0 defines a 0-dimensional variety in An × Q s n−1 n Ps−1 with D := i=1 di E⊂{s+1,...,r},#E=n−s j∈E dj ≤ s−1 d points lying in An × {µs 6= 0}, and the Jacobian determinant of the polynomials obtained from the gi ’s by setting µs = 1 does not vanish at any of these points. (ii) The variety defined in Pn × Ps−1 by the polynomials obtained homogenizing g1 , . . . , gr in the variables x with a new variable x0 is contained in {x0 6= 0} × Ps−1 . (iii) All the solutions to the system g1 = · · · = gr = 0 can be listed within O(Dn log2 (n)) arithmetic operations in Q. (iv) There is an slp of length O(dn2 ) which evaluates all the polynomials g1 , . . . , gr . Proof. Items (i) and (ii) are consequences of the non-singularity of Cauchy matrices. Then, (iii) follows by applying an algorithm to solve linear systems with Cauchy matrices (see [10, Ch. 2, Alg. 4.2]) and (iv) is immediate. 11
3.3.2
A geometric resolution
n Consider F1 , . . . , Fr as in Section 3.3.1 and let V (e) = V (F1 , . . . , Fr ) ⊂ AK(t) × s−1 . PK(t)
n Lemma 3.11 The variety V (e) ⊂ AK(t) ×{µs 6= 0} and has D points S1 , . . . , SD , r
which are in K[[t − 1]]r ⊂ K(t) .
Proof. The multihomogeneous B´ezout theorem (see, for instance, [36, Chapter 4, Section 2.1]) states that the degree of V (e) is bounded by D. The result follows applying the Newton-Hensel lifting to the solutions of the initial system g1 = 0, . . . , gr = 0 at t = 1 (see e.g. [20, Lemma 3]). Let y = (y1 , . . . , ynP ) be new variables and ℓ(x, µ, y) = α ∈ Cn , let ℓα (x, µ) = nj=1 αj xj . Let P (t, U, y) =
D Y i=1
Pn
j=1 yj xj .
For
Pˆ (t, U, y) U − ℓ(Si , y) = ∈ K(t)[U, y], q(t)
with q(t) monic and Pˆ (t, U, y) ∈ K[t][U, y] primitive with respect to t. In order to compute P we will approximate its roots. The required precision is obtained from the following upper bound for the degree of its coefficients. P h Lemma 3.12 If Pˆ (t, U, y) = D h=0 ph (t, y)U , then degt ph ≤ nD for every 0 ≤ h ≤ D. Proof. As in the proof of [22, Lemma 2.3], it follows that degt Pˆ is bounded by the number of isolated zeroes of F1 (t, x) = · · · = Fs (t, x) = Fs+1 (t, x, µ) = · · · = Fr (t, x, µ) = ℓ(β1 ,...,βn ) (x) − β0 = 0. By the multihomogeneous B´ezout theorem, this system has at most nD isolated solutions: each partition of {1, . . . , r} into sets Ex′ and Eµ′ of cardinality n and s − 1 respectively with Eµ′ ⊂ {s + 1, . . . , r} leads to n partitions of {1, . . . , r + 1} into sets Ex , Eµ and Et of cardinality n, s − 1 and 1 respectively with Eµ ⊂ {s + 1, . . . , r} and Et ⊂ {1, . . . , r}, by taking Ex = (Ex′ \ {e}) ∪ {r + 1}, Eµ = Eµ′ and Et = {e} for every e ∈ Ex′ , and every required partition of {1, . . . , r + 1} can be obtained in this way. In each case, the degree product corresponding to Ex , Eµ , Et is bounded by the corresponding to Ex′ , Eµ′ . 12
Q If π(V ∩ {t = 0}) = {z1 , . . . , zν } ⊂ An × Ps−1 , and aj=1 qj (U, y)δj is the factorization of Pˆ (0, U, y) in C[U, y], we may suppose ql = U − ℓ(zl , y) for Q δ −1 every 1 ≤ l ≤ ν ≤ a. If g(U, y) = aj=1 qj j , for a generic α ∈ Cn , n Pˆ (0, U, α) g(U, α)
;
∂ Pˆ (0, U, α) ∂U
g(U, α)
;−
∂ Pˆ (0, U, α) ∂yk
g(U, α)
o , k = 1, . . . , n
(2)
is a geometric resolution of a finite set containing πx (V ∩ {t = 0}), where πx : A × An × Ps−1 → An is the projection (t, x, µ) 7→ x. The algorithmic counterpart of this construction is the following: Proposition 3.13 There is a probabilistic algorithm that, taking as input polynomials f1 , . . . , fr in K[x, µ] as in Section 3.3.1 encoded by an slp of length L, obtains a geometric resolution as (2)within complexity O n2 D 2 log(D) log log(D) L + dn2 + log2 (D) log log(D) .
Proof. The procedure of this algorithm is standard. The difference in complexity with other algorithms (e.g. [17, 20, 34]) is obtained by doing the Newton lifting pointwise. First, for i = 1, . . . , D, the algorithm computes S˜i ∈ K[t]r such that for 1 ≤ k ≤ r, (S˜i − Si )k ∈ (t − 1)2nD+1 K[[t − 1]] by applying recursively the Newton-Hensel lifting to the roots of the initial system g1 , . . . , gr defined in Notation 3.9 within complexity O(n2 (L + dn2 )D 2 log(D) logP log(D)). h In the next step, the algorithm computes Pˆ (0, U, α) = D h=0 ph (0, α)U P D ∂ph ∂ Pˆ h n and ∂y (0, U, α) = h=0 ∂yk (0, α)U for a generic α = (α1 , . . . , αn ) ∈ Q k as follows: first, it computes of U h and U h (yk − αk ) (1 ≤ QD the coefficients k ≤ n, 0 ≤ h ≤ D) in i=1 (U − ℓ(S˜i , y)) mod (t − 1)2nD+1 following [40, Algorithm 10.3] in O(n2 D 2 log3 (D) log log2 (D)) operations over K. From h (t, α) (1 ≤ k ≤ n, 0 ≤ h ≤ D), these coefficients, the polynomials ph (t, α), ∂p ∂yk and q(t) are obtained within complexity O(n2 D 2 log2 (D) log log(D)) over K by using [40, Corollary 5.24 and Algorithm 11.4] and converting all rational fractions to a common denominator. Finally, it evaluates them at t = 0. ˆ In the last step, g(U, α) = gcd(Pˆ (0, U, α), ∂∂UP (0, U, α)) and the required exact divisions by g(U, α) are computed to obtain a geometric resolution. Remark 3.14 The algorithm in Proposition 3.13 can be adapted straightforwardly within the same complexity to handle the cases s = 1 and s = n (see (0) (1) Remark 3.5). We keep the notation VˆS = VS ∪ VS ∪ VS for the deformation variety. 13
4 4.1
Solving the problem Regular intersections
Throughout this section we assume that the polynomials f1 , . . . , fm ∈ K[x1 , . . . , xn ] meet the following condition: Hypothesis 4.1 For every S = {i1 , . . . , is } ⊂ {1, . . . , m} and every x ∈ Cn , if fi1 (x) = · · · = fis (x) = 0, then {∇fi1 (x), . . . , ∇fis (x)} is a linearly independent set. This implies that, for every 1 ≤ s ≤ n, the zero locus of any subfamily of s polynomials is either empty or a complex manifold of dimension n − s. Moreover, for s > n, a subfamily of s of the polynomials defines the empty set. As in Section 3.2, for S = {i1 , . . . , is } ⊂ {1, . . . , m}, consider the solution set WS ⊂ An × Ps−1 of the system (1). From Proposition 3.6 we deduce: Corollary 4.2 If C 6= ∅ is a connected component of {f1 = · · · = fq = S 0, fq+1 > 0, . . . , fm > 0} ⊂ Rn , then q ≤ n and Z(C) ⊂ Π {1,...,q}⊂S⊂{1,...,m} WS . 1≤#S≤n S If the conditions in Proposition 3.2 hold, Π S⊂{1,...,m}, 1≤#S≤n WS contains a point in the closure of each connected component C of the realization of every feasible sign condition over f1 , . . . , fm such that Π1 (C) 6= R. Lemma 4.3 After a generic linear change of variables, for every S ⊂ {1, . . . , m}, Π(WS ) is finite. Furthermore, if s ≤ n − 1, then WS is also a finite set. Proof. When s = n, Hypothesis 4.1 implies that Π(WS ) is finite. If s ≤ n−1, Π(WS ) is the set of critical points of the map x 7→ x1 on US = {x | fi (x) = 0 for i ∈ S}. By the arguments in [38, Section 2.1] based on Sard’s theorem and a holomorphic Morse lemma, a generic linear form has a finite number of critical points on US . Taking any of these linear forms as the first coordinate, Π(WS ) turns to be finite. As, for every x ∈ Π(WS ), {∇fi (x), i ∈ S} is linearly dependent and {∇fi (x), i ∈ S} is linearly independent, there is a unique (x, µ) ∈ WS . Then WS is a finite set. Therefore, to deal with our problem under Hypothesis 4.1 it is enough to solve polynomial systems of type (1). 14
Theorem 4.4 There is a probabilistic algorithm that, given f1 , . . . , fm ∈ K[x1 , . . . , xn ] of degrees bounded by d ≥ 2 satisfying Hypothesis 4.1 and encoded by an slp of length L, computes a finite family of geometric resolutions of 0-dimensional varieties whose union contains at least one point in the closure of each connected component of the realization of every sign condition Pmin{m,n} m n−12 2n 4 over f1 , . . . , fm within complexity O d n log(d)(log(n)+ s=1 s s−1 2 2 log log(d)) L + n + nd + n log(n) log (d) .
Proof. After an initial random linear change of variables, we may assume f1 , . . . , fm are in the situation described in Proposition 3.2 and the recursion starts. Firstly, by using Proposition 3.13, for every S ⊂ {1, . . . , m} of cardinality s ≤ n, the algorithm computes a geometric resolution of Π(WS ) = πx (VS ∩ {t = 0}) (see Corollary 3.8 and Lemma 4.3). By Corollary S 4.2, the geometric resolutions computed describe a finite set containing C∈C Z(C). Now, for a randomly chosen p ∈ Kn and every 2 ≤ k ≤ n, we may assume that fi (p1 , . . . , pk−1, xk , . . . , xn ), 1 ≤ i ≤ m, satisfy Hypothesis 4.1. Then, for k = 2, . . . , n − 1, the same procedure as before is applied to fi (p1 , . . . , pk−1 , xk , . . . , xn ), 1 ≤ i ≤ n, S to compute geometric resolutions of 0-dimensional varieties containing C∈C(k,p) Z(C, k). Finally, note that S Sm i=1 {fi (p1 , . . . , pn−1 , xn ) = 0}. C∈C(n,p) Z(C, n) ⊂ The output of the algorithm is the point p together with the family of geometric resolutions computed during the recursion (see Proposition 3.4). The complexity follows from the complexity estimate in Proposition 3.13. Now we show how to get the entire list of feasible sign conditions over f1 , . . . , fm satisfying Hypothesis 4.1. Notation 4.5 For every σ ∈ {}m with exactly t “=” coordinates, Pσ denotes the subset of {}m consisting of the 3t elements obtained from σ by replacing some of its “=” either with “”.
Proposition 4.6 Let L = {σ ∈ {}m | σ is feasible over f1 , . . . , fm }. Given a finite set M such that M ∩ C 6= ∅ for every connected component C of the realization of each σ ∈ L, let L(M) = {σ ′ ∈ {}m | ∃z ∈ S M satisfying σ ′ over f1 , . . . , fm }. Then L = σ′ ∈L(M) Pσ′ . 15
Proof. Let σ ′ ∈ L(M). Assume σ ′ = (=, . . . , =, >, . . . , >) with t “=”. If t = 0, Pσ′ = {σ ′ }. If t > 0, let σ ∈ Pσ′ . We may assume σ = (=, . . . , =, > , . . . , >) with q “=” where 0 ≤ q ≤ t. Let z ∈ M be such that fi (z)σi′ 0 for 1 ≤ i ≤ m. Since ∇f1 (z), . . . , ∇ft (z) are linearly independent, there exists v ∈ Rn such that h∇fi (z), vi = 0 for 1 ≤ i ≤ q and h∇fi (z), vi > 0 for q + 1 ≤ i ≤ t. Consider a regular curve γ : [−1, 1] → {f1 = · · · = fq = 0} such that γ(0) = z and γ ′ (0) = v. For q + 1 ≤ i ≤ m, we have that, for a sufficiently small and positive value u, fi ◦ γ(u) > 0 holds. As, for 1 ≤ i ≤ q, fi ◦ γ(u) = 0 for every u ∈ [−1, 1], σ ∈ L. The other inclusion is obvious using continuity. Theorem 4.7 There is a probabilistic algorithm that, given f1 , . . . , fm ∈ K[x1 , . . . , xn ] as in Theorem 4.4 computes the list of all feasible sign conditions over these polynomials from the output of the algorithm underlying that Pmin{m,n} m n−1Ω Ωn n−1 n d m + L d n log(d) theorem within complexity O s=1 s s−1 s−1 (log(n) + log log(d)) .
Proof. The output of the algorithm in Theorem 4.4 is, for every (k, S) with 1 ≤ k ≤ n and S ⊂ {1, . . . , m} such that s = #S ≤ k, a geometric reso(k,S) (k,S) lution {q (k,S) , w1 , . . . , wn } ⊂ K[U] consisting of polynomials of degrees bounded by D(k,S) ≤ k−1 dk . Then, for every (k, S), the algorithm computes s−1 (k,S)
(k,S)
fi (w1 (U), . . . , wn (U)) mod q (k,S) (U) (1 ≤ i ≤ m) within O(LD(k,S) log(D(k,S) ) log log(D(k,S))) operations and evaluates their signs at the zeroes of q (k,S) within complexity O(m(D(k,S) )Ω ) by applying the procedure in [13, Section 3]. Finally, the list of feasible sign conditions over f1 , . . . , fm is obtained as stated in Proposition 4.6.
If we are interested in a particular sign condition σ over f1 , . . . , fm with q equalities, by Corollary 4.2, our algorithm can suitably be adapted: (a) to obtain a finite family of geometric resolutions of 0-dimensional varieties intersecting the closure of each connected component of the realizaPmin{m,n} m−q n−12 2n 4 d n log(d)(log(n) + tion of σ within complexity O s=q+1 s−q s−1 2 2 log log(d)) L + n + nd + n log(n) log (d) ; Pmin{m,n} m−q n−1Ω Ωn (b) to decide the feasibility of σ within O d m+ s=q+1 s−q s−1 n−1 n L s−1 d n log(d)(log(n) + log log(d)) additional operations. 16
4.2
A single polynomial
Let f ∈ K[x1 , . . . , xn ]. Assume the conditions in Proposition 3.2 hold. Notation 4.8 Let d := deg f and d˜ := 2⌈ d2 ⌉. Let T be the Tchebychev ˜ Set g1 := n + 1 + Pn T (xk ), polynomial of the first kind of degree d. k=1 ∂f ∂g1 F := g1 + (t − 1)(g1 − f ), and, for 2 ≤ i ≤ n, fi := ∂x , g := = T ′ (xi ) i ∂xi i ∂F and Fi := gi + (t − 1)(gi − fi ) = ∂x . i Note that g1 > 0 over Rn , since T ≥ −1 over R (for properties of Tchebychev polynomials see, for instance, [23]), and then, F (t, x) 6= 0 for t 6= 0 and x ∈ {f = 0}. The system g1 = · · · = gn = 0 defines a 0-dimensional variety ˜ d˜ − 1)n−1 points, and its Jacobian determinant does not in An with D := d( vanish at any of these points. Let W = {f = f2 = · · · = fn = 0} ⊂ An and Vˆ = {F = F2 = · · · = Fn = 0} = V (0) ∪ V (1) ∪ V ⊂ A × An as before. Lemma 4.9 With our previous assumptions, for each connected component C of {f = 0}, {f > 0} or {f < 0}, Z(C) ⊂ π(V ∩ {t = 0}). Proof. Consider first a connected component C of {f = 0} (for a similar approach in this case with an alternative deformation, see [30].) Let y ∈ Zi (C). Take ε > 0 such that B(y, ε)∩{f = 0} ⊂ C and B(y, ε)∩Z(C) = {y}. There exists µ ∈ (y1 , y1 + ε) such that ∂B(y, ε) ∩ C ⊂ {x1 > µ}. Let K := ∂B(y, ε)∩{x1 ≤ µ}. Assume f > 0 over K. Let ε0 ∈ (0, ε) be such that F > 0 over [−ε0 , ε0 ] × K and let t1 ∈ (0, ε0). Take z ∈ B(y, ε) with z1 < y1 . As F (−t1 , y) < 0, F (t1 , y) > 0 and F (0, z) 6= 0, there is a point (t2 , z˜) in the union of the segments (−t1 , y)(0, z) and (0, z)(t1 , y) such that F (t2 , z˜) = 0 and t2 6= 0. Then z˜1 < y1 < µ and (t2 , z˜) ∈ {F = 0} ∩ ({t2 } × B(y, ε)). Let (t2 , w) be a point in this set where x1 attains its minimum. Then, √w ∈ B(y, ε) and so, if ε is small enough, (t2 , w) ∈ V . As |(t2 , w) − (0, y)| < 2ε and V is closed, (0, y) ∈ V . Now, assume C is a connected component of {f > 0} and let y ∈ Zi (C). Let C ′ be the connected component of {f = 0} containing y. Let ε > 0 such that B(y, ε) ∩ {f = 0} ⊂ C ′ and B(y, ε) ∩ Z(C) = {y}. There exists µ ∈ (y1 , y1 + ε) such that ∂B(y, ε) ∩ C ⊂ {x1 > µ}. Let γ : [0, 1] → Rn be a continuous semialgebraic curve with γ(0) = y and γ((0, 1]) ⊂ B(y, ε) ∩ C ∩ {x1 < µ}, and let C1 be the connected component of B(y, ε) ∩ C such that γ((0, 1]) ⊂ C1 . Take t1 ∈ (−ε, 0) so that F (t1 , γ(1)) > 0. There exists 17
u1 ∈ (0, 1) such that F (t1 , γ(u1)) = 0. Consider the connected component {t1 } × Ct′1 of {F = 0} ∩ ({t1 } × B(y, ε)) containing (t1 , γ(u1 )). Then, Ct′1 ∪ C1 is a connected set and, as f 6= 0 over Ct′1 , Ct′1 ⊂ C1 . Let K2 = Ct′1 ∪(B(y, ε)∩ {x1 ≥ µ}), which is compact. Let (t1 , w) ∈ K2 be a point where x1 attains its minimum. Then, w ∈ Ct′1 ⊂ B(y, ε) and (t1 , w) ∈ V if ε is small enough. √ Since |(t1 , w) − (0, y)| < 2ε, (0, y) ∈ V . The algorithm is an adaptation of the one in Theorem 4.4: Theorem 4.10 There is a probabilistic algorithm that, given a polynomial f ∈ K[x1 , . . . , xn ] of degree bounded by an even integer d˜ ≥ 2 encoded by an slp of length L, computes a finite family of geometric resolutions of 0dimensional varieties whose union contains at least one point in the closure of each connected component of the realization of every sign condition on f ˜ + nΩ )d˜2n log2 (d)(log(n) ˜ ˜ 2 ). within complexity O(n5 (L + dn + log log(d)) Proof. By Lemma 4.9 and Proposition 3.4, after a random linear change of variables, it is enough to construct a geometric resolution of a finite set containing π(V ∩ {t = 0}) and proceed recursively taking p = 0 ∈ Kn . ˜ If T˜ denotes the Tchebychev polynomial of the first kind of degree d/2, ′ ′ ′ ˜ then gcd(T + 1, T ) = T˜ and gcd(T Pn− 1, T ) = T /T . For E : {2, . . . , n} → {−1, 1}, let g1,E (x1 ) = T (x1 ) + ( k=2 E(k)) + n + 1 and, for 2 ≤ i ≤ n, gi,E (xi ) = T˜(xi ) if E(i) = −1 and gi,E (xi ) = (T ′ /T˜ )(xi ) if E(i) = 1. Since ′ T S takes the value 1 or −1 at any root of T , {g1 = g2 = · · · = gn = 0} = E {g1,E (x1 ) = g2,E (x2 ) = · · · = gn,E (xn ) = 0}, where the union runs over all E : {2, . . . , n} → {−1, 1}. n Let Pnα ∈ Q be generic, I := (y1 − α1 , . . . , yn − αn ) ⊂ K[y1 , . . . , yn ] and ℓ = i=1 yi xi . Since g1,E Q, . . . , gn,E are polynomials in separate variables, we compute QE (U, y) := gi,E (s)=0 (U − ℓ(s, y)) ∈ Q[U, y]/I 2 , proceeding as in [22, Section 5.2.1], with O(nDE2 log2 (DE ) log log(DE )) operations, where Q DE = d˜ ni=2 deg gi,E . Finally, the product of all the QE (U, y) is computed applying [40, Algorithm 10.3] within complexity O(nD 2 log2 (D) log log(D)) and a geometric resolution of {g1 = · · · = gn = 0} is obtained. Now, the algorithm computes P (t, U, y) mod (t − 1)2nD+1 by recursive application of [17, Algorithm 1]. After performing this computation, the procedure runs as the algorithm in Proposition 3.13. The complexity of the whole step is O(n3 (L + dn + nΩ )D 2 log2 (D) log log2 (D)). The overall complexity of the algorithm follows by adding the complexities of all the recursive steps. 18
4.3
A sketch of the bivariate case
Here, we show that, when n = 2, the algorithm underlying the proof of Theorem 4.4 solves the problem for an arbitrary finite family of polynomials. According to Remark 3.5, the polynomial systems considered do not involve µ variables and all the considered varieties are affine. We keep the notation introduced in Section3.3. For lack of space, we only state our results and the ideas involved. Q Lemma 4.11 Let f1 , f2 ∈ C[x1 , x2 ] be nonzero polynomials, and f1 = ai=1 pδi i Q ǫ and f2 = bj=1 qj j their irreducible factorizations in C[x1 , x2 ]. Assume that p1 and q1 are not associates. Then, for every z ∈ C2 such that p1 (z) = q1 (z) = 0 and pi (z) 6= 0, qj (z) 6= 0 for i, j ≥ 2, we have z ∈ π(V{1,2} ∩ {t = 0}). Lemma 4.12 Let f1 ∈ C[x1 , x2 ] be a nonzero polynomial without factors in Q C[x1 ]\ C and relatively prime to g1 (see Notation 3.9). Assume f1 = ai=1 pδi i is its irreducible factorization in C[x1 , x2 ]. Let z ∈ C2 satisfying that either ∂p there is an index i0 such that pi0 (z) = ∂xi20 (z) = 0 or there are indices i1 6= i2 such that pi1 (z) = pi2 (z) = 0. Then, z ∈ π(V{1} ∩ {t = 0}). Proof. The relies on the fact that ((F1 , F2 ) : t∞ ) = P proof Q Q ∂pi (F1 , F2 , g1 i δi ∂x ( j6=i pj ) − g2 i pi ). Then, (0, z) ∈ V ((F1 , F2 ) : t∞ ) and 2 therefore z ∈ π(V{1} ∩ {t = 0}). Proposition 4.13 Let f1 , . . . , fm ∈ K[x1 , x2 ] be arbitrary polynomials. After change of variables, S a generic linear S C∈C Z(C) ⊂ S⊂{1,...,m}, 1≤#S≤2 π(VS ∩ {t = 0}).
Proof. A generic linear change of variables ensures that, for each C ∈ C, either Π1 (C) = R or Z(C) is non-empty and finite (see Proposition 3.2) and each fi is relatively prime to its corresponding g. If z ∈ Z(C), by applying the implicit function theorem to those irreducible factors of f1 , . . . , fm vanishing at z, the hypotheses of either Lemma 4.11 or Lemma 4.12 hold and, then, z ∈ π(VS ) for some S with 1 ≤ #S ≤ 2. The output of this algorithm enables us to list all feasible closed sign conditions σ ∈ {≤, =, ≥}m within the same complexity as in Theorem 4.7.
19
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[28] J. Renegar. On the computational complexity and geometry of the first-order theory of the reals. I, II, III. J. Symbolic Comput., 13(3):255–352, 1992. [29] F. Rouillier. Solving zero-dimensional systems through the rational univariate representation. Appl. Algebra Engrg. Comm. Comput., 9(5):433–461, 1999. [30] F. Rouillier, M.-F. Roy, and M. Safey El Din. Finding at least one point in each connected component of a real algebraic set defined by a single equation. J. Complexity, 16(4):716–750, 2000. [31] M. Safey El Din. Testing sign conditions on a multivariate polynomial and applications. To appear in Mathematics in Computer Science. ´ Schost. Polar varieties and computation of one point [32] M. Safey El Din and E. in each connected component of a smooth algebraic set. In Proc. of ISSAC 2003, pages 224–231, New York, 2003. ACM. [33] M. Safey El Din and P. Tr´ebuchet. Strong bi-homogeneous b´ezout theorem and its use in effective real algebraic geometry. Manuscript. ´ Schost. Computing parametric geometric resolutions. Appl. Algebra Engrg. [34] E. Comm. Comput., 13(5):349–393, 2003. [35] A. Seidenberg. A new decision method for elementary algebra. Ann. of Math. (2), 60:365–374, 1954. [36] I. R. Shafarevich. Basic algebraic geometry. Springer-Verlag, Berlin, 1977. [37] A. Tarski. A decision method for elementary algebra and geometry. U. of California Press, Berkeley, 1951. [38] C. Voisin. Hodge theory and complex algebraic geometry II. Cambridge Univ. Press, Cambridge, 2003. [39] J. von zur Gathen. Parallel arithmetic computations: a survey. In Mathematical foundations of computer science, volume 233 of Lect. Notes Comput. Sci., pages 93–112. Springer, Berlin, 1986. [40] J. von zur Gathen and J. Gerhard. Modern computer algebra. Cambridge University Press, New York, 1999.
22
On sign conditions over real multivariate polynomials Gabriela Jeronimo∗† Daniel Perrucci∗
Juan Sabia∗
Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina
Abstract We present a new probabilistic algorithm to find a finite set of points intersecting the closure of each connected component of the realization of every sign condition over a family of real polynomials defining regular hypersurfaces that intersect transversally. This enables us to show a probabilistic procedure to list all feasible sign conditions. We extend our main algorithm to the case of an arbitrary multivariate polynomial and to that of an arbitrary family of bivariate polynomials. The complexity bounds for these procedures improve the known ones.
Keywords: Real multivariate polynomials, sign conditions and consistency problem, algorithms and complexity.
1
Introduction
Given f1 , . . . , fm ∈ R[x1 , . . . , xn ], a sign condition σ ∈ {}m is said to be feasible if the system f1 σ1 0, . . . , fm σm 0 has a solution in Rn , and the set of solutions is called the realization of σ. One of the basic problems in ∗ Partially supported by the following Argentinian research grants: UBACyT X112 (2004-2007), UBACyT X847 (2006-2009), CONICET PIP 5852/05 † Partially supported by ANPCYT PICT 2005 17–33018.
1
computational semialgebraic geometry is to decide whether a sign condition is feasible. This problem is a particular case of quantifier elimination and, on the other hand, many elimination algorithms use subroutines deciding all the feasible sign conditions for a family of polynomials. The first elimination algorithms over the reals are due to Tarski [37] and Seidenberg [35] but their complexity is not elementary recursive. Collins [14] was the first to obtain a doubly exponential complexity. In [18], Grigor’ev and Vorobjov presented an algorithm with single exponential complexity to decide the consistency of systems of inequalities by studying the critical points of a function. This idea was also used to obtain more efficient quantifier elimination procedures ([21, 28, 7]). The algorithms rely on the computation of a finite set of points intersecting every connected component of a semialgebraic set. A standard technique was to take sums of squares and introduce infinitesimals to study only smooth and compact hypersurfaces. The specific problem of consistency for equalities over R was treated through the critical point method afterwards. In [30], the case of a single equation is studied, reducing the introduction of infinitesimals, and in [2], an algorithm with no infinitesimals is given to deal with arbitrary positive dimensional systems. Several probabilistic procedures lead to successive improvements. Using classical polar varieties, in [3, 4], the case of a smooth compact variety given by a regular sequence is tackled within a complexity depending polynomially on an intrinsic degree of the systems involved and the input length. To achieve this complexity, straight-line programs and an efficient procedure to solve polynomial equation systems over C ([15]) are used. The compactness assumption is dropped in [5, 6] by using generalized polar varieties. The non-compact case is also considered in [32] for a smooth equidimensional variety defined by a radical ideal by studying projections over polar varieties, and an extension to the non-equidimensional situation is given in [33]. Finally, [31] deals with sets of the type {f > 0} through the computation of generalized critical points. In this paper, we present a probabilistic algorithm that, given polynomials, obtains a parametric description of a finite set of points intersecting the closure of each connected component of the realization of any sign condition over them in three cases: firstly, when the polynomials define regular hypersurfaces in Cn intersecting transversally (see Hypothesis 4.1); secondly, for a single arbitrary multivariate polynomial, and finally, for a family of arbitrary bivariate polynomials. In the first case, the output of the algorithm allows to obtain all the feasible sign conditions by using the techniques in [13]. 2
The input and intermediate computations in our algorithms are encoded by straight-line programs (see Section 2.2). The output is described by means of geometric resolutions, that is to say, by univariate rational parametrizations of 0-dimensional varieties (see Section 2.3). The sketch of the algorithm is the following: given polynomials f1 , . . . , fm , a generic change of variables ensures that each connected component C ⊂ Rn of each feasible sign condition over them avoids some asymptotic behavior with respect to the projection on the first coordinate (see Section 3.1). More precisely, either the projection of C on x1 is R or the projection of C is a proper closed interval. In this last case, points in C are obtained as extremal points of x1 . These points are solutions of particular systems of equations (see Section 3.2) which are dealt with by means of deformation techniques (see Section 3.3) to obtain geometric resolutions of finite sets including them. When the projection is R, C is intersected with x1 = p1 for some value p1 and the algorithm follows recursively. The following theorem summarizes our main results: Theorem Let K be a subfield of R. There is a probabilistic algorithm that, from f1 , . . . , fm ∈ K[x1 , . . . , xn ] defining regular hypersurfaces in Cn which intersect transversally, with degrees bounded by d ≥ 2 and encoded by a straight-line program of length L, computes a finite family of geometric resolutions of 0-dimensional varieties whose union contains at least one point in the closure of each connected component of the realization of every sign conPmin{m,n} m n−1 n 2 dition over f1 , . . . , fm within O d (L + d) operations s=1 s s−1 in K up to logarithmic factors. The same result holds in the case of a single polynomial without any hypothesis and of an arbitrary family of bivariate polynomials. n The factor n−1 d in the complexity is just an upper bound for the s−1 bihomogeneous B´ezout numbers (see [36]) of the systems arising from the Lagrange characterization of critical points of a map (cf. [33]). One of the new tools to achieve our complexity order, which improves the previous ones depending on these parameters, is the use of algorithmic deformation techniques specially designed for bihomogeneous systems (see [19] for a similar approach). Up to now, these systems were handled with general algorithms for solving complex polynomial equations (see, for instance, [1, 29, 17, 26]). Our work can also be seen as an extension of [32] and [6] in the sense that we deal not only with equations but also with inequalities. In the three situations previously mentioned, our algorithm may be adapted 3
to deal with a particular sign condition within a lower complexity order. Moreover, for polynomials defining regular hypersurfaces which intersect transversally, we also give a procedure for computing the list of all feasible sign conditions over them from the output of our main algorithm with Pmin{m,n} m n−1 n Ω n−1 n O d m + d L additional operations (up to s=1 s s−1 s−1 logarithmic factors) where Ω > 2 is a number such that two ℓ × ℓ matrices can be multiplied with O(ℓΩ ) operations. For a treatment of the same problem in full generality, see for instance [8]. As in the case of bivariate polynomials, we expect our deformation approach will work in the general multivariate setting without any assumptions. This is subject of our current research. This paper is organized as follows: In Section 2 we state some preliminary notions. Section 3 is devoted to presenting the basic tools used in the design of our algorithms: first, we show some properties of generic changes of variables; then, we present the Lagrange-type polynomial equation systems characterizing critical points, and finally, the algorithmic deformation techniques for bihomogeneous systems are shown. In Section 4, we present our main algorithms.
2 2.1
Preliminaries Notation
Throughout this paper Q, R and C denote the fields of rational, real and complex numbers respectively and K denotes an effective subfield of R. If k is a field, k¯ denotes an algebraic closure of k. If A is a set in a topological space, A denotes its closure. For a positive integer n and an algebraically closed field k, we denote by Ank and Pnk (or by An or Pn if k is clear from the context) the n-dimensional affine and projective space over k respectively, equipped with their Zariski topologies. We adopt the usual notion of dimension and degree for algebraic varieties (see for instance [36]). Let Π1 : Rn → R be the projection on the first coordinate. For any nonempty set A ⊂ Rn , we define Zi (A) = {(x1 , . . . , xn ) ∈ A | x1 = inf Π1 (A)} if Π1 (A) is bounded from below, and Zi(A) = ∅ otherwise. Similarly, Zs (A) = {(x1 , . . . , xn ) ∈ A | x1 = sup Π1 (A)} if Π1 (A) is bounded from above, and Zs (A) = ∅ otherwise. Finally, we denote Z(A) = Zi (A) ∪ Zs (A). Given f1 , . . . , fm ∈ R[x1 , . . . , xn ], C will denote the set of all the connected 4
components of the realizations of all feasible sign conditions σ ∈ {}m over f1 , . . . , fm .
2.2
Algorithms and complexity
The algorithms we consider are described by arithmetic networks over K (see [39]). The complexity of an algorithm is the number of operations and comparisons in K. Our algorithms are probabilistic in the sense that they depend on some random choices of points and their output is correct whenever these points lie outside proper Zariski closed sets. The objects we deal with are polynomials with coefficients in K. Each polynomial is represented either as the array of all its coefficients in a prefixed order of all the monomials, or by a straight-line program. Roughly speaking, a straight-line program (or slp, for short) over K encoding a list of polynomials in K[x1 , . . . , xn ] is an arithmetic circuit which enables us to evaluate these polynomials at any given point in Kn . The number of instructions is called the length of the slp (for a precise definition we refer to [12, Definition 4.2]).
2.3
Geometric resolutions
A way of representing 0-dimensional affine varieties that is widely used in computer algebra nowadays is a geometric resolution. This notion, which was first introduced by Kronecker [25] and K¨onig [24], appears in the literature under different names such as rational univariate representation, shape lemma, etc. (for a historical account, see [17]). The precise definition we use is the following: Let k be a field and V = {ξ (1) , . . . , ξ (D) } a 0-dimensional variety in Ank¯ defined by polynomials in k[x1 , . . . , xn ]. Given a separating linear form ℓ ∈ k[x1 , . . . , xn ] for V (that is, ℓ(ξ (i) ) 6= ℓ(ξ (j) ) if i 6= j), V is characterized by: the Qif U is a new variable, (i) minimal polynomial p := 1≤i≤D (U − ℓ(ξ )) ∈ k[U] of ℓ over V , a polynomial p˜ ∈ k[U] with deg(˜ p) < D and relatively prime to p, and polynomials w1 , . . . , wn ∈ k[U] with deg(wj ) < D for every 1 ≤ j ≤ n satisfying n w o wn ¯ n 1 ¯ p(η) = 0 . V = (η), . . . , (η) ∈ k | η ∈ k, p˜ p˜
The family {p, p˜, w1 , . . . , wn } ⊂ k[U] is called a geometric resolution of V (associated with ℓ). As p˜ is invertible in k[U]/(p(U)), this is equivalent 5
to the standard notion a family {p, v1 , . . . , vn } in of geometric resolution: n ¯ p(η) = 0 . We will k[U] satisfying V = v1 (η), . . . , vn (η) ∈ k | η ∈ k, use both definitions alternatively, since the complexity of passing from one representation to the other does not modify the overall complexity of our algorithms.
3
General approach
3.1
Avoiding asymptotic situations
Notation 3.1 For 1 ≤ k ≤ n, Πk : Rn → R is the projection Πk (x1 , . . . , xn ) = xk . For A ⊂ Rn , Zi (A, k) = {(x1 , . . . , xn ) ∈ A | xk = inf Πk (A)} if A is bounded from below and Zi (A, k) = ∅ otherwise. Similarly, Zs (A, k) = {(x1 , . . . , xn ) ∈ A | xk = sup Πk (A)} if A is bounded from above and Zi (A, k) = ∅ otherwise. Finally, Z(A, k) = Zi (A, k)∪Zs (A, k). In particular, Zi (A, 1) = Zi (A), Zs (A, 1) = Zi (A) and Z(A, 1) = Z(A) (see Section 2.1). Given f1 , . . . , fm ∈ R[x1 , . . . , xn ] and p ∈ Rn , C(k, p) is the set of all the connected components of the Rn -subsets C ∩ {x1 = p1 , . . . , xk−1 = pk−1 } with C ∈ C. In particular, C(1, p) = C for every p ∈ Rn (see Section 2.1). The non-asymptotic behavior for every step of the algorithm ensured by a generic change of variables is stated in: Proposition 3.2 Let f1 , . . . , fm ∈ R[x1 , . . . , xn ]. Then, after a generic change of variables over Q, for every p ∈ Rn , 1 ≤ k ≤ n and C ∈ C(k, p), Z(C, k) is finite (possibly empty), and if Πk (C) is bounded from below or above, then Z(C, k) 6= ∅. To prove Proposition 3.2, we will use the following: Lemma 3.3 Let {fij }1≤i≤n,1≤j≤li ⊂ R[x1 , . . . , xn ] be a family of nonzero polynomials satisfying simultaneously: a) for 1 ≤ i ≤ n, {fij }1≤j≤li ⊂ R[x1 , . . . , xi ] is closed under derivation with respect to xi and every fij is quasi-monic (that is, monic up to a constant) in xi , b) for 1 ≤ i ≤ n, there exists 1 ≤ li′ ≤ li such that for 1 < i ≤ n, ′ {f(i−1)j }1≤j≤li−1 slices {fij }1≤j≤li in the sense of [11, Definition 2.3.4]. 6
Let p ∈ Rn and let 1 ≤ k ≤ i ≤ n. Let D ⊂ Ri be defined by a boolean formula on the polynomials fij , 1 ≤ j ≤ li′ , involving equalities and inequalities to zero, and let C be a connected component of D ∩ {x1 = p1 , . . . , xk−1 = pk−1 }. Then the set Z(C, k) is finite (possibly empty). Moreover, if Πk (C) is bounded from below or above, then Z(C, k) 6= ∅. Proof. As {fij (p1 , . . . , pk−1 , xk , . . . , xn )}k≤i≤n,1≤j≤li in R[xk , . . . , xn ] is a family that satisfies the hypotheses (1 ≤ k ≤ n), it is enough to prove the lemma for k = 1. For i = 1, the result is clear. Suppose the statement is true for i − 1. Let Π : Ri → Ri−1 be the projection on the first i − 1 coordinates. Following the notation in [11, Ch. 2], let A1 , . . . , Aℓ be the slicing of Ri−1 with respect to fi1 , . . . , fili ′ and, for 1 ≤ s ≤ ℓ, let ξs,1 < · · · < ξs,as : given by f(i−1)1 , . . . , f(i−1)li−1 As → R be the continuous semialgebraic functions that slice As × R. Let ′ , . . . , f(i−1)li−1 As,1 , . . . , As,us be the connected S components of As . As f(i−1)1S slice fi1 , . . . , fili′ , Π(C) = h Ash ,uh , and then Z(Π(C)) ⊂ h Z(Ash ,uh ). Since each Ash can be described by a boolean formula only involving the ′ , by inductive hypothesis, Z(Π(C)) is finite. polynomials f(i−1)1 , . . . , f(i−1)li−1 Now, if w ∈ Z(C), Π(w) ∈ Z(Π(C)). Moreover, fi1 , . . . , fili′ are quasi-monic in xi and at least one of them vanishes at w. Therefore, Z(C) is a finite set. Suppose now that Π1 (C) is an interval bounded, for example, from below. Then, there exists z = (z1 , . . . , zi−1 ) ∈ Zi (Π(C)) ⊂ Π(C). Let γ : [0, 1] → Ri−1 be a continuous semialgebraic curve such that γ((0, 1]) ⊂ Π(C) and γ(0) = z (see [11, Theorem 2.5.5]). We may suppose that for 0 < ε ≤ 1, γ((0, ε]) ⊂ A1 . Using [11, Lema 2.5.6], each ξ1,a can be extended to a semialgebraic continuous function ξ 1,a over A1 . Let x˜ := γ(ε) and y ∈ R such that (˜ x, y) ∈ C. Depending on the position of y with respect to ξ 1,1 (˜ x) < x), it is easy in any case to define a continuous semialgebraic · · · < ξ 1,a1 (˜ function h : [0, ε] → R such that γ˜ : [0, ε] → Ri defined as γ˜ (t) = (γ(t), h(t)) satisfies γ˜ ((0, ε]) ⊂ C (note that the signs of fi1 , . . . , fili are constant over γ˜ ((0, ε])) and, therefore, (z, h(0)) = γ˜ (0) ∈ C. As z1 = inf Π1 (Π(C)) = inf Π1 (C), (z, h(0)) ∈ Zi (C). Now, we can prove Proposition 3.2: Proof. Let V := (vr,s )1≤r,s≤n be a matrix whose entries are new variables and consider the family of polynomials {Fij }1≤i≤n,1≤j≤li ⊂ R[v, x] defined in the following way: 7
• Take ln′ = m and, for 1 ≤ j ≤ ln′ , let Fnj (V, x) = fj (V x). Then, for 1 ≤ j0 ≤ ln′ , if degx Fnj0 = dnj0 , add its first dnj0 − 1 derivatives with respect to xn to the list to obtain the family {Fnj }1≤j≤ln . • From {F(i0 +1)j }1≤j≤li0 +1 ⊂ R[v, x1 , . . . , xi0 +1 ], first, take all possible resultants and subresultants with respect to xi0 +1 between pairs of elements to form {Fi0 j }1≤j≤li′ ⊂ R[v, x1 , . . . , xi0 ], not considering the ones that are iden0 tically zero. Then, for 1 ≤ j0 ≤ li′0 , if degx Fi0 j0 = di0 j0 , add its first di0 j0 − 1 derivatives with respect to the variable xi0 to obtain the family {Fi0 j }1≤j≤li0 . From the recursive definition of {Fij }1≤i≤n,1≤j≤li , it can be shown induc A 0 tively that for 1 ≤ j ≤ li′ and A ∈ Qi×i , Fij (V, Ax) = Fij V ,x , 0 In−i B0 B 0 ′ (i−1)×(i−1) and for li +1 ≤ j ≤ li and B ∈ Q , Fij V, x = Fij V ,x . 0 1
0 In−i+1 d
Using these identities, if dij := degx Fij and qij ∈ R[v] is the coefficient of xi ij in Fij (1 ≤ i ≤ n, 1 ≤ j ≤ li ), the existence of a linear change of variables making Fij ′ (1 ≤ j ′ ≤ li′ ) quasi-monic in xi implies that qij 6≡ 0. Let U = {V0 ∈ Cn×n | qij (V0 ) 6= 0 for 1 ≤ i ≤ n, 1 ≤ j ≤ li }. By [9, Proposition 4.34 and Theorem 5.14], for every V0 ∈ Qn×n ∩ U, the set {fij (x)}1≤i≤n,1≤j≤li defined by fij (x) = Fij (V0 , x) satisfies both conditions in Lemma 3.3 and fnj = fj (V0 x) for 1 ≤ j ≤ ln′ . The result follows.
The following proposition is a major tool for our algorithm (cf. [32, Theorem 2]). n Proposition , after a generic change of variables, the Sn 3.4S For every p ∈ R set {p} ∪ k=1 C∈C(k,p) Z(C, k) is finite and has at least one point in the closure of each connected component of the realization of each feasible sign condition over the polynomials f1 , . . . , fm .
Proof. By Proposition 3.2 the set is finite. Let C ∈ C. Then, either Z(C) is a finite non-empty set and intersects C or Π1 (C) = R (see again Proposition 3.2). Now, it suffices to consider the connected components of C ∩ {x1 = p1 } and the projection Π2 . The result follows recursively. This is the recursion in our algorithm. Note that, for 2 ≤ k ≤ n, the k-th variable can be seen as the first one for fj (p1 , . . . , pk−1, xk , . . . , xn ) (1 ≤ j ≤ m). Therefore, S it suffices to consider the problem of finding a finite set which contains C∈C Z(C). 8
3.2
Equations defining extremal points
Let f1 , . . . , fm ∈ R[x1 , . . . , xn ] and let S := {i1 , . . . , is } ⊂ {1, . . . , m}. If 1 ≤ s < n, the implicit function theorem implies that the points with maximum or minimum first coordinate x1 in a connected component of {fi1 = · · · = fis = 0} satisfy ( fi1 (x) = · · · = fis (x) = 0, Ps (1) n−1 j=1 µj ∇fij (x) = (0, . . . , 0) ∈ R
for µ1 , . . . , µs ∈ R not simultaneously zero, where ∇fij (x) denotes the vector obtained by removing the first coordinate from the gradient vector ∇fij (x). System (1) can be considered for any s ≤ m and any {i1 , . . . , is } ⊂ {1, . . . , m} and defines a variety WS ⊂ AnC ×PCs−1 . Let Π denote the projection Π : AnC × PCs−1 → AnC .
Remark 3.5 If s ≥ n or s = 1, Π(WS ) is the solution set of fi1 (x) = · · · = ∂f ∂f fis (x) = 0 or fi1 (x) = ∂xi21 (x) = · · · = ∂xin1 (x) = 0 respectively. We will characterize Π(WS ) by means of these simpler systems in these cases. The following result is an adaptation of the Karush-Kuhn-Tucker conditions for non-linear optimization (see [27, Chapter 3]) which generalize the Lagrange multipliers theorem: Proposition 3.6 If C is a connected componentSof the set {f1 = · · · = fq = 0, fq+1 > 0, . . . , fm > 0} ⊂ Rn , then Z(C) ⊂ Π {1,...,q}⊂S⊂{1,...,m}, WS . S6=∅
Proof. Let z = (z1 , . . . , zn ) ∈ Zi (C). Note that {i ∈ {1, . . . , m} | fi (z) = 0} is not empty and, without loss of generality, assume it equals {1, . . . , t} with q ≤ t ≤ m. Suppose that {∇fi (z), 1 ≤ i ≤ t} is linearly independent. Let f = (f1 , . . . , ft ) : Rn → Rt . We may suppose that the minor corresponding to the variables n − t + 1, . . . , n in the Jacobian matrix Df (z) is not zero. Applying the inverse function theorem to h(x) = (x1 − z1 , . . . , xn−t − zn−t , f1 (x), . . . , ft (x)), there exist an open neighborhood U of z, ε ∈ R>0 and a map g : (−ε, ε)n → U inverse to h : U → (−ε, ε)n . Let w ∈ C ∩ U and y = (y1 , . . . , yn ) = h(w). Let γ : [−ε/2, y1 ] → (−ε, ε)n be γ(u) = (u, y2, . . . , yn ). Then, Im(g ◦ γ) is a connected curve and g ◦ γ(y1 ) = w ∈ C. On the other hand, for 1 ≤ i ≤ q, fi (g ◦ γ(u)) = 0 and for q + 1 ≤ i ≤ t, fi (g ◦ γ(u)) > 0. This implies that Im(g ◦ γ) ⊂ {f1 = 9
· · · = fq = 0, fq+1 > 0, . . . , fm > 0} (for t + 1 ≤ i ≤ m, fi (z) > 0 and therefore we may suppose fi > 0 over U). Thus, Im(g ◦ γ) ⊂ C, but the first coordinate of g ◦ γ(−ε/2) is −ε/2 + z1 < z1 . This contradicts the fact that z1 = inf Π1 (C). Therefore, {∇fi (z), 1 ≤ i ≤ t} should be linearly dependent and z ∈ Π(W{1,...,t} ).
3.3
Deformation techniques for bihomogeneous systems
In this section we describe a procedure based on the symbolic deformation techniques developed in [16, 15, 17, 20, 34], adapted to the bihomogeneous setting following [19], that we will apply to deal with the specific polynomial systems of type (1). 3.3.1
The deformation
We consider a system of equations given by polynomials in K[x1 , . . . , xn , µ1 , . . . , µs ]: f1 (x) = 0, . . . , fs (x) = 0, fs+1 (x, µ) = 0, . . . , fr (x, µ) = 0 where 2 ≤ s ≤ n − 1 and r = s + n − 1, such that, for 1 ≤ i ≤ r, degx fi := di ≤ d and, for s+1 ≤ i ≤ r, fi is homogeneous of degree 1 in the variables µ. (To avoid introducing notation, we call these polynomials f1 , . . . , fr ; however, they will not be the input polynomials of our algorithms but new polynomials obtained as in (1)). Let W ⊂ AnC × PCs−1 be the variety they define. Let g1 (x), . . . , gs (x), gs+1(x, µ), . . . , gr (x, µ) ∈ K[x, µ] with the same degrees and homogeneity as f1 , . . . , fr such that they define a 0-dimensional variety in An × Ps−1 and their homogenizations in x with a new variable x0 define a variety in Pn × Ps−1 which is contained in {x0 6= 0}. For every 1 ≤ i ≤ r, let Fi := gi + (t − 1)(gi − fi ). Consider the variety Vˆ ⊂ A×An ×Ps−1 defined by F1 , . . . , Fr , and write Vˆ = V (0) ∪V (1) ∪V, where V (0) is the union of the irreducible components of Vˆ contained in {t = 0}, V (1) is the union of those contained in {t = t0 } for some t0 ∈ C \ {0}, and V is the union of the remaining ones. In Section 4, given a family of m polynomials in K[x], for S ⊂ {1, . . . , m}, we will add a subscript S in the notation Vˆ , V (0) , V (1) and V to indicate that these varieties are defined from the bihomogeneous system (1) associated with S. Lemma 3.7 Under the previous hypotheses, Vˆ ∩ {t = 1} = V ∩ {t = 1} and every irreducible component of V is 1-dimensional and intersects {t = 1}. 10
Proof. Each irreducible component of Vˆ has dimension at least 1 and Vˆ ∩ {t = 1} is 0-dimensional. Therefore, Vˆ ∩ {t = 1} = V ∩ {t = 1}. Let V1 be an irreducible component of V and V1 its Zariski closure in 1 A × Pn × Ps−1 . The projection of V1 to A1 is onto and so, V1 ∩ {t = 1} = 6 ∅. But our assumption on g1 , . . . , gr implies that V1 ∩ {t = 1} = V1 ∩ {t = 1}. Then V1 ∩ {t = 1} = 6 ∅ and it is 0-dimensional. It follows that dim(V1 ) = 1. The procedure we are going to apply will enable us to compute a geometric resolution of a finite subset of the variety Π(W ) defined by our input system. Let π : A × An × Ps−1 → An × Ps−1 be the projection (t, x, µ) 7→ (x, µ). Corollary 3.8 With our previous assumptions, π(V ∩ {t = 0}) is a finite subset of W containing its isolated points. In what follows, we introduce some polynomials g1 , . . . , gr we will deal with and show some of their basic properties. Qi (xi − j). For s + 1 ≤ i ≤ r, for Notation 3.9 For 1 ≤ i ≤ s, gi (x) := dj=1 Pn 1 1 1 ≤ j ≤ di , if φ(i,j)(x) := k=s+1 (i−s−1)d+j−1+k−s xk + (i−s−1)d+j−1+n+1−s , Ps Qdi 1 then gi (x, µ) := k=1 i−s−1+k µk . j=1 φ(i,j) (x)
Lemma 3.10 Let g1 , . . . , gr be as in Notation 3.9. Then: (i) The system Q g1 = · ·· =Pgr = 0 defines a 0-dimensional variety in An × Q s n−1 n Ps−1 with D := i=1 di E⊂{s+1,...,r},#E=n−s j∈E dj ≤ s−1 d points lying in An × {µs 6= 0}, and the Jacobian determinant of the polynomials obtained from the gi ’s by setting µs = 1 does not vanish at any of these points. (ii) The variety defined in Pn × Ps−1 by the polynomials obtained homogenizing g1 , . . . , gr in the variables x with a new variable x0 is contained in {x0 6= 0} × Ps−1 . (iii) All the solutions to the system g1 = · · · = gr = 0 can be listed within O(Dn log2 (n)) arithmetic operations in Q. (iv) There is an slp of length O(dn2 ) which evaluates all the polynomials g1 , . . . , gr . Proof. Items (i) and (ii) are consequences of the non-singularity of Cauchy matrices. Then, (iii) follows by applying an algorithm to solve linear systems with Cauchy matrices (see [10, Ch. 2, Alg. 4.2]) and (iv) is immediate. 11
3.3.2
A geometric resolution
n Consider F1 , . . . , Fr as in Section 3.3.1 and let V (e) = V (F1 , . . . , Fr ) ⊂ AK(t) × s−1 . PK(t)
n Lemma 3.11 The variety V (e) ⊂ AK(t) ×{µs 6= 0} and has D points S1 , . . . , SD , r
which are in K[[t − 1]]r ⊂ K(t) .
Proof. The multihomogeneous B´ezout theorem (see, for instance, [36, Chapter 4, Section 2.1]) states that the degree of V (e) is bounded by D. The result follows applying the Newton-Hensel lifting to the solutions of the initial system g1 = 0, . . . , gr = 0 at t = 1 (see e.g. [20, Lemma 3]). Let y = (y1 , . . . , ynP ) be new variables and ℓ(x, µ, y) = α ∈ Cn , let ℓα (x, µ) = nj=1 αj xj . Let P (t, U, y) =
D Y i=1
Pn
j=1 yj xj .
For
Pˆ (t, U, y) U − ℓ(Si , y) = ∈ K(t)[U, y], q(t)
with q(t) monic and Pˆ (t, U, y) ∈ K[t][U, y] primitive with respect to t. In order to compute P we will approximate its roots. The required precision is obtained from the following upper bound for the degree of its coefficients. P h Lemma 3.12 If Pˆ (t, U, y) = D h=0 ph (t, y)U , then degt ph ≤ nD for every 0 ≤ h ≤ D. Proof. As in the proof of [22, Lemma 2.3], it follows that degt Pˆ is bounded by the number of isolated zeroes of F1 (t, x) = · · · = Fs (t, x) = Fs+1 (t, x, µ) = · · · = Fr (t, x, µ) = ℓ(β1 ,...,βn ) (x) − β0 = 0. By the multihomogeneous B´ezout theorem, this system has at most nD isolated solutions: each partition of {1, . . . , r} into sets Ex′ and Eµ′ of cardinality n and s − 1 respectively with Eµ′ ⊂ {s + 1, . . . , r} leads to n partitions of {1, . . . , r + 1} into sets Ex , Eµ and Et of cardinality n, s − 1 and 1 respectively with Eµ ⊂ {s + 1, . . . , r} and Et ⊂ {1, . . . , r}, by taking Ex = (Ex′ \ {e}) ∪ {r + 1}, Eµ = Eµ′ and Et = {e} for every e ∈ Ex′ , and every required partition of {1, . . . , r + 1} can be obtained in this way. In each case, the degree product corresponding to Ex , Eµ , Et is bounded by the corresponding to Ex′ , Eµ′ . 12
Q If π(V ∩ {t = 0}) = {z1 , . . . , zν } ⊂ An × Ps−1 , and aj=1 qj (U, y)δj is the factorization of Pˆ (0, U, y) in C[U, y], we may suppose ql = U − ℓ(zl , y) for Q δ −1 every 1 ≤ l ≤ ν ≤ a. If g(U, y) = aj=1 qj j , for a generic α ∈ Cn , n Pˆ (0, U, α) g(U, α)
;
∂ Pˆ (0, U, α) ∂U
g(U, α)
;−
∂ Pˆ (0, U, α) ∂yk
g(U, α)
o , k = 1, . . . , n
(2)
is a geometric resolution of a finite set containing πx (V ∩ {t = 0}), where πx : A × An × Ps−1 → An is the projection (t, x, µ) 7→ x. The algorithmic counterpart of this construction is the following: Proposition 3.13 There is a probabilistic algorithm that, taking as input polynomials f1 , . . . , fr in K[x, µ] as in Section 3.3.1 encoded by an slp of length L, obtains a geometric resolution as (2)within complexity O n2 D 2 log(D) log log(D) L + dn2 + log2 (D) log log(D) .
Proof. The procedure of this algorithm is standard. The difference in complexity with other algorithms (e.g. [17, 20, 34]) is obtained by doing the Newton lifting pointwise. First, for i = 1, . . . , D, the algorithm computes S˜i ∈ K[t]r such that for 1 ≤ k ≤ r, (S˜i − Si )k ∈ (t − 1)2nD+1 K[[t − 1]] by applying recursively the Newton-Hensel lifting to the roots of the initial system g1 , . . . , gr defined in Notation 3.9 within complexity O(n2 (L + dn2 )D 2 log(D) logP log(D)). h In the next step, the algorithm computes Pˆ (0, U, α) = D h=0 ph (0, α)U P D ∂ph ∂ Pˆ h n and ∂y (0, U, α) = h=0 ∂yk (0, α)U for a generic α = (α1 , . . . , αn ) ∈ Q k as follows: first, it computes of U h and U h (yk − αk ) (1 ≤ QD the coefficients k ≤ n, 0 ≤ h ≤ D) in i=1 (U − ℓ(S˜i , y)) mod (t − 1)2nD+1 following [40, Algorithm 10.3] in O(n2 D 2 log3 (D) log log2 (D)) operations over K. From h (t, α) (1 ≤ k ≤ n, 0 ≤ h ≤ D), these coefficients, the polynomials ph (t, α), ∂p ∂yk and q(t) are obtained within complexity O(n2 D 2 log2 (D) log log(D)) over K by using [40, Corollary 5.24 and Algorithm 11.4] and converting all rational fractions to a common denominator. Finally, it evaluates them at t = 0. ˆ In the last step, g(U, α) = gcd(Pˆ (0, U, α), ∂∂UP (0, U, α)) and the required exact divisions by g(U, α) are computed to obtain a geometric resolution. Remark 3.14 The algorithm in Proposition 3.13 can be adapted straightforwardly within the same complexity to handle the cases s = 1 and s = n (see (0) (1) Remark 3.5). We keep the notation VˆS = VS ∪ VS ∪ VS for the deformation variety. 13
4 4.1
Solving the problem Regular intersections
Throughout this section we assume that the polynomials f1 , . . . , fm ∈ K[x1 , . . . , xn ] meet the following condition: Hypothesis 4.1 For every S = {i1 , . . . , is } ⊂ {1, . . . , m} and every x ∈ Cn , if fi1 (x) = · · · = fis (x) = 0, then {∇fi1 (x), . . . , ∇fis (x)} is a linearly independent set. This implies that, for every 1 ≤ s ≤ n, the zero locus of any subfamily of s polynomials is either empty or a complex manifold of dimension n − s. Moreover, for s > n, a subfamily of s of the polynomials defines the empty set. As in Section 3.2, for S = {i1 , . . . , is } ⊂ {1, . . . , m}, consider the solution set WS ⊂ An × Ps−1 of the system (1). From Proposition 3.6 we deduce: Corollary 4.2 If C 6= ∅ is a connected component of {f1 = · · · = fq = S 0, fq+1 > 0, . . . , fm > 0} ⊂ Rn , then q ≤ n and Z(C) ⊂ Π {1,...,q}⊂S⊂{1,...,m} WS . 1≤#S≤n S If the conditions in Proposition 3.2 hold, Π S⊂{1,...,m}, 1≤#S≤n WS contains a point in the closure of each connected component C of the realization of every feasible sign condition over f1 , . . . , fm such that Π1 (C) 6= R. Lemma 4.3 After a generic linear change of variables, for every S ⊂ {1, . . . , m}, Π(WS ) is finite. Furthermore, if s ≤ n − 1, then WS is also a finite set. Proof. When s = n, Hypothesis 4.1 implies that Π(WS ) is finite. If s ≤ n−1, Π(WS ) is the set of critical points of the map x 7→ x1 on US = {x | fi (x) = 0 for i ∈ S}. By the arguments in [38, Section 2.1] based on Sard’s theorem and a holomorphic Morse lemma, a generic linear form has a finite number of critical points on US . Taking any of these linear forms as the first coordinate, Π(WS ) turns to be finite. As, for every x ∈ Π(WS ), {∇fi (x), i ∈ S} is linearly dependent and {∇fi (x), i ∈ S} is linearly independent, there is a unique (x, µ) ∈ WS . Then WS is a finite set. Therefore, to deal with our problem under Hypothesis 4.1 it is enough to solve polynomial systems of type (1). 14
Theorem 4.4 There is a probabilistic algorithm that, given f1 , . . . , fm ∈ K[x1 , . . . , xn ] of degrees bounded by d ≥ 2 satisfying Hypothesis 4.1 and encoded by an slp of length L, computes a finite family of geometric resolutions of 0-dimensional varieties whose union contains at least one point in the closure of each connected component of the realization of every sign condition Pmin{m,n} m n−12 2n 4 over f1 , . . . , fm within complexity O d n log(d)(log(n)+ s=1 s s−1 2 2 log log(d)) L + n + nd + n log(n) log (d) .
Proof. After an initial random linear change of variables, we may assume f1 , . . . , fm are in the situation described in Proposition 3.2 and the recursion starts. Firstly, by using Proposition 3.13, for every S ⊂ {1, . . . , m} of cardinality s ≤ n, the algorithm computes a geometric resolution of Π(WS ) = πx (VS ∩ {t = 0}) (see Corollary 3.8 and Lemma 4.3). By Corollary S 4.2, the geometric resolutions computed describe a finite set containing C∈C Z(C). Now, for a randomly chosen p ∈ Kn and every 2 ≤ k ≤ n, we may assume that fi (p1 , . . . , pk−1, xk , . . . , xn ), 1 ≤ i ≤ m, satisfy Hypothesis 4.1. Then, for k = 2, . . . , n − 1, the same procedure as before is applied to fi (p1 , . . . , pk−1 , xk , . . . , xn ), 1 ≤ i ≤ n, S to compute geometric resolutions of 0-dimensional varieties containing C∈C(k,p) Z(C, k). Finally, note that S Sm i=1 {fi (p1 , . . . , pn−1 , xn ) = 0}. C∈C(n,p) Z(C, n) ⊂ The output of the algorithm is the point p together with the family of geometric resolutions computed during the recursion (see Proposition 3.4). The complexity follows from the complexity estimate in Proposition 3.13. Now we show how to get the entire list of feasible sign conditions over f1 , . . . , fm satisfying Hypothesis 4.1. Notation 4.5 For every σ ∈ {}m with exactly t “=” coordinates, Pσ denotes the subset of {}m consisting of the 3t elements obtained from σ by replacing some of its “=” either with “”.
Proposition 4.6 Let L = {σ ∈ {}m | σ is feasible over f1 , . . . , fm }. Given a finite set M such that M ∩ C 6= ∅ for every connected component C of the realization of each σ ∈ L, let L(M) = {σ ′ ∈ {}m | ∃z ∈ S M satisfying σ ′ over f1 , . . . , fm }. Then L = σ′ ∈L(M) Pσ′ . 15
Proof. Let σ ′ ∈ L(M). Assume σ ′ = (=, . . . , =, >, . . . , >) with t “=”. If t = 0, Pσ′ = {σ ′ }. If t > 0, let σ ∈ Pσ′ . We may assume σ = (=, . . . , =, > , . . . , >) with q “=” where 0 ≤ q ≤ t. Let z ∈ M be such that fi (z)σi′ 0 for 1 ≤ i ≤ m. Since ∇f1 (z), . . . , ∇ft (z) are linearly independent, there exists v ∈ Rn such that h∇fi (z), vi = 0 for 1 ≤ i ≤ q and h∇fi (z), vi > 0 for q + 1 ≤ i ≤ t. Consider a regular curve γ : [−1, 1] → {f1 = · · · = fq = 0} such that γ(0) = z and γ ′ (0) = v. For q + 1 ≤ i ≤ m, we have that, for a sufficiently small and positive value u, fi ◦ γ(u) > 0 holds. As, for 1 ≤ i ≤ q, fi ◦ γ(u) = 0 for every u ∈ [−1, 1], σ ∈ L. The other inclusion is obvious using continuity. Theorem 4.7 There is a probabilistic algorithm that, given f1 , . . . , fm ∈ K[x1 , . . . , xn ] as in Theorem 4.4 computes the list of all feasible sign conditions over these polynomials from the output of the algorithm underlying that Pmin{m,n} m n−1Ω Ωn n−1 n d m + L d n log(d) theorem within complexity O s=1 s s−1 s−1 (log(n) + log log(d)) .
Proof. The output of the algorithm in Theorem 4.4 is, for every (k, S) with 1 ≤ k ≤ n and S ⊂ {1, . . . , m} such that s = #S ≤ k, a geometric reso(k,S) (k,S) lution {q (k,S) , w1 , . . . , wn } ⊂ K[U] consisting of polynomials of degrees bounded by D(k,S) ≤ k−1 dk . Then, for every (k, S), the algorithm computes s−1 (k,S)
(k,S)
fi (w1 (U), . . . , wn (U)) mod q (k,S) (U) (1 ≤ i ≤ m) within O(LD(k,S) log(D(k,S) ) log log(D(k,S))) operations and evaluates their signs at the zeroes of q (k,S) within complexity O(m(D(k,S) )Ω ) by applying the procedure in [13, Section 3]. Finally, the list of feasible sign conditions over f1 , . . . , fm is obtained as stated in Proposition 4.6.
If we are interested in a particular sign condition σ over f1 , . . . , fm with q equalities, by Corollary 4.2, our algorithm can suitably be adapted: (a) to obtain a finite family of geometric resolutions of 0-dimensional varieties intersecting the closure of each connected component of the realizaPmin{m,n} m−q n−12 2n 4 d n log(d)(log(n) + tion of σ within complexity O s=q+1 s−q s−1 2 2 log log(d)) L + n + nd + n log(n) log (d) ; Pmin{m,n} m−q n−1Ω Ωn (b) to decide the feasibility of σ within O d m+ s=q+1 s−q s−1 n−1 n L s−1 d n log(d)(log(n) + log log(d)) additional operations. 16
4.2
A single polynomial
Let f ∈ K[x1 , . . . , xn ]. Assume the conditions in Proposition 3.2 hold. Notation 4.8 Let d := deg f and d˜ := 2⌈ d2 ⌉. Let T be the Tchebychev ˜ Set g1 := n + 1 + Pn T (xk ), polynomial of the first kind of degree d. k=1 ∂f ∂g1 F := g1 + (t − 1)(g1 − f ), and, for 2 ≤ i ≤ n, fi := ∂x , g := = T ′ (xi ) i ∂xi i ∂F and Fi := gi + (t − 1)(gi − fi ) = ∂x . i Note that g1 > 0 over Rn , since T ≥ −1 over R (for properties of Tchebychev polynomials see, for instance, [23]), and then, F (t, x) 6= 0 for t 6= 0 and x ∈ {f = 0}. The system g1 = · · · = gn = 0 defines a 0-dimensional variety ˜ d˜ − 1)n−1 points, and its Jacobian determinant does not in An with D := d( vanish at any of these points. Let W = {f = f2 = · · · = fn = 0} ⊂ An and Vˆ = {F = F2 = · · · = Fn = 0} = V (0) ∪ V (1) ∪ V ⊂ A × An as before. Lemma 4.9 With our previous assumptions, for each connected component C of {f = 0}, {f > 0} or {f < 0}, Z(C) ⊂ π(V ∩ {t = 0}). Proof. Consider first a connected component C of {f = 0} (for a similar approach in this case with an alternative deformation, see [30].) Let y ∈ Zi (C). Take ε > 0 such that B(y, ε)∩{f = 0} ⊂ C and B(y, ε)∩Z(C) = {y}. There exists µ ∈ (y1 , y1 + ε) such that ∂B(y, ε) ∩ C ⊂ {x1 > µ}. Let K := ∂B(y, ε)∩{x1 ≤ µ}. Assume f > 0 over K. Let ε0 ∈ (0, ε) be such that F > 0 over [−ε0 , ε0 ] × K and let t1 ∈ (0, ε0). Take z ∈ B(y, ε) with z1 < y1 . As F (−t1 , y) < 0, F (t1 , y) > 0 and F (0, z) 6= 0, there is a point (t2 , z˜) in the union of the segments (−t1 , y)(0, z) and (0, z)(t1 , y) such that F (t2 , z˜) = 0 and t2 6= 0. Then z˜1 < y1 < µ and (t2 , z˜) ∈ {F = 0} ∩ ({t2 } × B(y, ε)). Let (t2 , w) be a point in this set where x1 attains its minimum. Then, √w ∈ B(y, ε) and so, if ε is small enough, (t2 , w) ∈ V . As |(t2 , w) − (0, y)| < 2ε and V is closed, (0, y) ∈ V . Now, assume C is a connected component of {f > 0} and let y ∈ Zi (C). Let C ′ be the connected component of {f = 0} containing y. Let ε > 0 such that B(y, ε) ∩ {f = 0} ⊂ C ′ and B(y, ε) ∩ Z(C) = {y}. There exists µ ∈ (y1 , y1 + ε) such that ∂B(y, ε) ∩ C ⊂ {x1 > µ}. Let γ : [0, 1] → Rn be a continuous semialgebraic curve with γ(0) = y and γ((0, 1]) ⊂ B(y, ε) ∩ C ∩ {x1 < µ}, and let C1 be the connected component of B(y, ε) ∩ C such that γ((0, 1]) ⊂ C1 . Take t1 ∈ (−ε, 0) so that F (t1 , γ(1)) > 0. There exists 17
u1 ∈ (0, 1) such that F (t1 , γ(u1)) = 0. Consider the connected component {t1 } × Ct′1 of {F = 0} ∩ ({t1 } × B(y, ε)) containing (t1 , γ(u1 )). Then, Ct′1 ∪ C1 is a connected set and, as f 6= 0 over Ct′1 , Ct′1 ⊂ C1 . Let K2 = Ct′1 ∪(B(y, ε)∩ {x1 ≥ µ}), which is compact. Let (t1 , w) ∈ K2 be a point where x1 attains its minimum. Then, w ∈ Ct′1 ⊂ B(y, ε) and (t1 , w) ∈ V if ε is small enough. √ Since |(t1 , w) − (0, y)| < 2ε, (0, y) ∈ V . The algorithm is an adaptation of the one in Theorem 4.4: Theorem 4.10 There is a probabilistic algorithm that, given a polynomial f ∈ K[x1 , . . . , xn ] of degree bounded by an even integer d˜ ≥ 2 encoded by an slp of length L, computes a finite family of geometric resolutions of 0dimensional varieties whose union contains at least one point in the closure of each connected component of the realization of every sign condition on f ˜ + nΩ )d˜2n log2 (d)(log(n) ˜ ˜ 2 ). within complexity O(n5 (L + dn + log log(d)) Proof. By Lemma 4.9 and Proposition 3.4, after a random linear change of variables, it is enough to construct a geometric resolution of a finite set containing π(V ∩ {t = 0}) and proceed recursively taking p = 0 ∈ Kn . ˜ If T˜ denotes the Tchebychev polynomial of the first kind of degree d/2, ′ ′ ′ ˜ then gcd(T + 1, T ) = T˜ and gcd(T Pn− 1, T ) = T /T . For E : {2, . . . , n} → {−1, 1}, let g1,E (x1 ) = T (x1 ) + ( k=2 E(k)) + n + 1 and, for 2 ≤ i ≤ n, gi,E (xi ) = T˜(xi ) if E(i) = −1 and gi,E (xi ) = (T ′ /T˜ )(xi ) if E(i) = 1. Since ′ T S takes the value 1 or −1 at any root of T , {g1 = g2 = · · · = gn = 0} = E {g1,E (x1 ) = g2,E (x2 ) = · · · = gn,E (xn ) = 0}, where the union runs over all E : {2, . . . , n} → {−1, 1}. n Let Pnα ∈ Q be generic, I := (y1 − α1 , . . . , yn − αn ) ⊂ K[y1 , . . . , yn ] and ℓ = i=1 yi xi . Since g1,E Q, . . . , gn,E are polynomials in separate variables, we compute QE (U, y) := gi,E (s)=0 (U − ℓ(s, y)) ∈ Q[U, y]/I 2 , proceeding as in [22, Section 5.2.1], with O(nDE2 log2 (DE ) log log(DE )) operations, where Q DE = d˜ ni=2 deg gi,E . Finally, the product of all the QE (U, y) is computed applying [40, Algorithm 10.3] within complexity O(nD 2 log2 (D) log log(D)) and a geometric resolution of {g1 = · · · = gn = 0} is obtained. Now, the algorithm computes P (t, U, y) mod (t − 1)2nD+1 by recursive application of [17, Algorithm 1]. After performing this computation, the procedure runs as the algorithm in Proposition 3.13. The complexity of the whole step is O(n3 (L + dn + nΩ )D 2 log2 (D) log log2 (D)). The overall complexity of the algorithm follows by adding the complexities of all the recursive steps. 18
4.3
A sketch of the bivariate case
Here, we show that, when n = 2, the algorithm underlying the proof of Theorem 4.4 solves the problem for an arbitrary finite family of polynomials. According to Remark 3.5, the polynomial systems considered do not involve µ variables and all the considered varieties are affine. We keep the notation introduced in Section3.3. For lack of space, we only state our results and the ideas involved. Q Lemma 4.11 Let f1 , f2 ∈ C[x1 , x2 ] be nonzero polynomials, and f1 = ai=1 pδi i Q ǫ and f2 = bj=1 qj j their irreducible factorizations in C[x1 , x2 ]. Assume that p1 and q1 are not associates. Then, for every z ∈ C2 such that p1 (z) = q1 (z) = 0 and pi (z) 6= 0, qj (z) 6= 0 for i, j ≥ 2, we have z ∈ π(V{1,2} ∩ {t = 0}). Lemma 4.12 Let f1 ∈ C[x1 , x2 ] be a nonzero polynomial without factors in Q C[x1 ]\ C and relatively prime to g1 (see Notation 3.9). Assume f1 = ai=1 pδi i is its irreducible factorization in C[x1 , x2 ]. Let z ∈ C2 satisfying that either ∂p there is an index i0 such that pi0 (z) = ∂xi20 (z) = 0 or there are indices i1 6= i2 such that pi1 (z) = pi2 (z) = 0. Then, z ∈ π(V{1} ∩ {t = 0}). Proof. The relies on the fact that ((F1 , F2 ) : t∞ ) = P proof Q Q ∂pi (F1 , F2 , g1 i δi ∂x ( j6=i pj ) − g2 i pi ). Then, (0, z) ∈ V ((F1 , F2 ) : t∞ ) and 2 therefore z ∈ π(V{1} ∩ {t = 0}). Proposition 4.13 Let f1 , . . . , fm ∈ K[x1 , x2 ] be arbitrary polynomials. After change of variables, S a generic linear S C∈C Z(C) ⊂ S⊂{1,...,m}, 1≤#S≤2 π(VS ∩ {t = 0}).
Proof. A generic linear change of variables ensures that, for each C ∈ C, either Π1 (C) = R or Z(C) is non-empty and finite (see Proposition 3.2) and each fi is relatively prime to its corresponding g. If z ∈ Z(C), by applying the implicit function theorem to those irreducible factors of f1 , . . . , fm vanishing at z, the hypotheses of either Lemma 4.11 or Lemma 4.12 hold and, then, z ∈ π(VS ) for some S with 1 ≤ #S ≤ 2. The output of this algorithm enables us to list all feasible closed sign conditions σ ∈ {≤, =, ≥}m within the same complexity as in Theorem 4.7.
19
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