## Computing Parametric Geometric Resolutions (2001)

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Citations: | 20 - 7 self |

### BibTeX

@MISC{Schost01computingparametric,

author = {Eric Schost},

title = {Computing Parametric Geometric Resolutions},

year = {2001}

}

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### Abstract

Given a polynomial system of n equations in n unknowns that depends on some parameters, we de ne the notion of parametric geometric resolution as a means to represent some generic solutions in terms of the parameters. The coefficients of this resolution are rational functions of the parameters; we first show that their degree is bounded by the Bézout number d n , where d is a bound on the degrees of the input system. We then present a probabilistic algorithm to compute such a resolution; in short, its complexity is polynomial in the size of the output and the probability of success is controlled by a quantity polynomial in the Bézout number. We present several applications of this process, to computations in the Jacobian of hyperelliptic curves and to questions of real geometry.

### Citations

1241 | Commutative Algebra with a View Toward Algebraic Geometry - Eisenbud - 1995 |

392 | Fast Probabilistic Algorithms for Verification of Polynomial Identities
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Citation Context ...Subsection 4.3.2; this hypersurface has degree at most 4(n + 1)d 2n (4d n + 1) 2 . 33sEstimation of probabilities. We now return to the assumptions made on (p, p ′ , γ) and use Zippel-Schwartz’ lemma =-=[67, 60]-=- to quantify the choices that assure success. Let Γ be a subset of k, and suppose that the values (p, p ′ , γ) are chosen in Γ m × Γ m × Γ m−1 . Besides, we recall that the polynomial ∆ has degree at ... |

346 |
Gröbner Bases: an Algorithmic Method in Polynomial Ideal Theory. Recent trends in multidimensional systems theory
- Buchberger
- 1985
(Show Context)
Citation Context ...gorithm. For the resolution of zero-dimensional systems, we mention in particular the algorithm of geometric resolution [27, 26, 25, 28, 35]. Other approaches include the computation of Gröbner bases =-=[9, 21]-=-, possibly followed by a Rational Univariate Representation [50]. We also mention the linear algebra methods, using the matrices introduced by Macaulay [44] or generalizations thereof [19, 47, 18]. 6s... |

248 | A New Efficient Algorithm for Computing Gröbner Basis without Reduction to Zero: F5
- Faugère
- 2002
(Show Context)
Citation Context ...nalysis is done for a similar algorithm in [35]. Let us mention other approaches to this question, which were already presented in the introduction. Popular methods rely on Gröbner bases computations =-=[21, 20]-=-, either for a lexicographic ordering, or followed by the computation of a Rational Univariate Representation [50]. Other approaches include the computation of u-resultants by means of linear algebra ... |

235 |
Probabilistic algorithms for sparse polynomials
- Zippel
- 1979
(Show Context)
Citation Context ...Subsection 4.3.2; this hypersurface has degree at most 4(n + 1)d 2n (4d n + 1) 2 . 33sEstimation of probabilities. We now return to the assumptions made on (p, p ′ , γ) and use Zippel-Schwartz’ lemma =-=[67, 60]-=- to quantify the choices that assure success. Let Γ be a subset of k, and suppose that the values (p, p ′ , γ) are chosen in Γ m × Γ m × Γ m−1 . Besides, we recall that the polynomial ∆ has degree at ... |

181 |
Algebraic Complexity Theory
- Bürgisser, Claussen, et al.
- 1997
(Show Context)
Citation Context ...intrinsic complexity of the geometric objects V and π using the following notations. • The polynomials (f1, . . . , fn) are of degree bounded by d, and given by a Straight-Line Program of size L (see =-=[11]-=- for a definition). 8s• We call deg π the generic cardinality of the fibers of the restriction of π to V, that is the generic number of simple solutions of the specialized systems f(p, .) = 0. We call... |

173 |
On the computational complexity and geometry of the first-order theory of the reals
- Renegar
- 1992
(Show Context)
Citation Context ... the polynomial Qg(p, U) ∈ k[U] vanishes on the values taken by g in the fiber π −1 (p) ∩ V. We use this remark in Subsection 3.4. 14s3.3 Degree of a parametric resolution Using well-known techniques =-=[39, 44, 48, 4]-=-, we can recover a parametric resolution through the computation of the minimal polynomial of a “generic primitive element” of K → B; this minimal polynomial is also called a u-resultant [12] or a Cho... |

170 |
Elliptic curves over finite fields and the computation of square roots mod p
- Schoof
- 1985
(Show Context)
Citation Context ...on of the number of points of the Jacobian of a curve of genus 2 defined over the finite field k = Fp, where p is the first prime greater than 10 19 . Following Schoof’s algorithm for elliptic curves =-=[57]-=-, their algorithm is based on explicit computation of divisors of ℓ i -torsion, for various primes ℓ. This part describes the computations for the case ℓ = 2. We have considered a polynomial system wi... |

155 |
Efficient Computation of ZeroDimensional Gröbner Bases by Change of Ordering
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- 1993
(Show Context)
Citation Context ...gorithm. For the resolution of zero-dimensional systems, we mention in particular the algorithm of geometric resolution [27, 26, 25, 28, 35]. Other approaches include the computation of Gröbner bases =-=[9, 21]-=-, possibly followed by a Rational Univariate Representation [50]. We also mention the linear algebra methods, using the matrices introduced by Macaulay [44] or generalizations thereof [19, 47, 18]. 6s... |

146 |
Modern Computer Algebra
- Gathen, Gerhard
- 2003
(Show Context)
Citation Context ... expansion of p/q at precision 2D + 1. Given r as a polynomial of degree 2D, we want to compute p/q. In the single variable case, i.e. when m = 1, this question is solved using Padé approximants, see =-=[64]-=-. In our general multivariate case, the question can be solved using linear algebra; other solutions based on Gröbner bases computations are presented in [54, 22]. We propose an algorithm with better ... |

141 | Gröbner bases
- Becker, Weispfenning
- 1993
(Show Context)
Citation Context ...ive an exhaustive description of the solutions, describing all possibilities of degeneracy. We mention in particular the techniques of dynamic evaluation [16, 29, 14], the comprehensive Gröbner bases =-=[65]-=- and the computation of parametric Gröbner bases proposed in [32] and [45]. Whereas the complexities of the dynamic evaluation method or of Montes’ algorithm are not known to us, the approaches of Gri... |

140 |
The Algebraic Theory of Modular Systems
- Macaulay
- 1916
(Show Context)
Citation Context ...include the computation of Gröbner bases [9, 21], possibly followed by a Rational Univariate Representation [50]. We also mention the linear algebra methods, using the matrices introduced by Macaulay =-=[44]-=- or generalizations thereof [19, 47, 18]. 6sThe complexity of these zero-dimensional solving methods is not always known in terms of operations in the base field, which is here the rational function f... |

124 |
The complexity of partial derivatives
- Baur, Strassen
- 1983
(Show Context)
Citation Context ...ectors and matrices involved, and in the linear algebra operations. Evaluating the non-zero terms in the matrix Jac(Tκ) takes O(n deg π Ms(2 κ+1 , m)) operations in k. Using Baur-Strassen’s algorithm =-=[6]-=-, evaluating the matrix Jac(Fκ) and the vector Fκ takes O(nL) operations in the quotient Hκ. All linear algebra takes O(n 4 ) operations in Hκ, using for instance Leverrier’s algorithm [43] for matrix... |

101 |
Computing representations for the radicals of a finitely generated differential ideals
- Boulier, Lazard, et al.
(Show Context)
Citation Context .... The corresponding variety V = V(J ) ⊂ A m+n (k) is our object of interest. Let us denote by π : A m+n (k) → A m (k) the canonical projection on the parameter space A m (k). Then Lazard’s Lemma (see =-=[8]-=- and [46, Proposition 3.2]) shows the following fact: Fact 1 The restriction of π to each irreducible component of V(J ) is a dominant map, not necessarily finite but with generically finite fibers. T... |

97 |
Solving zero-dimensional systems through the rational univariate representation. Applicable Algebra
- Rouillier
- 1999
(Show Context)
Citation Context ... in particular the algorithm of geometric resolution [27, 26, 25, 28, 35]. Other approaches include the computation of Gröbner bases [9, 21], possibly followed by a Rational Univariate Representation =-=[50]-=-. We also mention the linear algebra methods, using the matrices introduced by Macaulay [44] or generalizations thereof [19, 47, 18]. 6sThe complexity of these zero-dimensional solving methods is not ... |

80 | A Gröbner free alternative for polynomial system solving
- Giusti, Lecerf, et al.
(Show Context)
Citation Context ...r computing such a parametric resolution, and work out a precise control of the probabilistic aspects. This algorithm is valid over any effective field, non necessarily perfect, as was often the case =-=[53, 34, 28]-=-. The complexity is polynomial in the size of the output. • As intermediate results, we give an algorithm for the rational reconstruction of a multivariate rational function. We also extend the Newton... |

70 |
Definability and fast quantifier elimination in algebraically closed fields
- Heintz
- 1983
(Show Context)
Citation Context ...ction of π to V, that is the generic number of simple solutions of the specialized systems f(p, .) = 0. We call deg V the degree of the variety V, using the notion of affine degree given by Heintz in =-=[33]-=-. Using the Bézout inequality of [33], both deg π and deg V can be bounded by the Bézout number d n . The complexities of our algorithms will be measured using the following notations. • The notation ... |

65 |
Grundzüge einer arithmetischen Theorie der algebraischen Grössen
- Kronecker
- 1882
(Show Context)
Citation Context ...ge and Strassen [56] and Schönhage [55]. • Ms(D, m) denotes the cost of m-variate series multiplication at precision D. This can be taken less than Mu((2D + 1) m ) using Kronecker’s substitution, see =-=[39]-=- and [64, ex. 16.16]. � �� ��� D+m If the base field k has characteristic zero, this complexity is in Olog Mu , i.e. m linear in the size of the series, up to logarithmic factors; see [42]. We make th... |

64 |
Exact solution of linear equations using p−adic expansions
- Dixon
- 1982
(Show Context)
Citation Context ...fting of the parameters presented here, and is used in practice. All necessary algorithmic tools are given in this paper, except the reconstruction of rational numbers, which is a well-solved problem =-=[62, 15, 64]-=-. Still, we do not give more details on this question: such a strategy induces a variety of new possibilities of failure, whose analysis requires to use arithmetic versions of Bézout’s theorem and of ... |

60 | Lower bounds for diophantine approximation
- Giusti, Hägele, et al.
- 1997
(Show Context)
Citation Context ...m). The idea of applying lifting techniques to solve polynomial systems can be traced back to the articles of Trinks [63] and Winkler [66]. It also underlies much of the recent work of the TERA group =-=[27, 26, 25, 28, 34, 35]-=-, where the use of the Straight-Line Program encoding was the key to algorithms with good complexity. 5sAs in the above references, we suppose that the input system is given by a StraightLine Program.... |

60 | When polynomial equation systems can be solved fast
- Giusti, Heintz, et al.
- 1995
(Show Context)
Citation Context ...ients that appear in a parametric resolution have degree bounded by an intrinsic geometric quantity, which itself is bounded by the Bézout number of the input system. This result is the continuity of =-=[53, 27]-=- and notably [34], and improves on the following important aspect. The results obtained in the above references require the zero-set of the defining system f = (f1, . . . , fn) to be in Noether positi... |

58 | Counting Points on Hyperelliptic Curves over Finite Fields
- Gaudry, Harley
- 2000
(Show Context)
Citation Context ...cobian amounts to solve a parametric polynomial system, whose parameters are the coordinates of the dividend. A parametric resolution of such a system was used in the genus 2 point-counting record of =-=[23]-=-. • Real geometry. As a particular case of parametric system, we may consider systems with infinitesimal coefficients, seeing the infinitesimals as parameters. Solving such systems is a cornerstone of... |

57 |
Schnelle Multiplikation von Polynomen über Körpern der Charakteristik 2
- Schönhage
- 1977
(Show Context)
Citation Context ... univariate polynomials of degree D, in terms of operations in the base ring. Mu(D) can be taken in O(D log D log log D), or Olog(D), using the algorithms of Schönhage and Strassen [56] and Schönhage =-=[55]-=-. • Ms(D, m) denotes the cost of m-variate series multiplication at precision D. This can be taken less than Mu((2D + 1) m ) using Kronecker’s substitution, see [39] and [64, ex. 16.16]. � �� ��� D+m ... |

56 | Straight–line programs in geometric elimination theory
- Giusti, Heintz, et al.
- 1998
(Show Context)
Citation Context .... In characteristic zero, using fast arithmetic, this complexity is linear, up to logarithmic factors, in the size of the output. Our use of the formal Newton operator is in the continuity of notably =-=[27, 26, 25, 28, 34, 35]-=-. In particular, a ready-to-implement formal Newton operator was given in the article [28]. Our method generalizes this algorithm to a wider class of representations, triangular sets representations. ... |

53 | Zeroes, multiplicities and idempotents for zerodimensional systems
- Alonso, Becker, et al.
- 1996
(Show Context)
Citation Context ... in: • a primitive element u = � uixi of K → K[X1, . . . , Xn]/J, • its monic minimal polynomial Qu ∈ K[U], • a parametrization of the algebraic variables in terms of the primitive element. Following =-=[4, 50, 28]-=-, we use in priority a parametrization of the form Q ′ u(u)xi = Vi(u) in K[X1, . . . , Xn]/J, where Vi is in K[U], for i = 1, . . . , n. In our particular context, we call parametric geometric resolut... |

49 | Algebraic solution of systems of polynomial equations using Gröbner bases. In Applied algebra, algebraic algorithms and error-correcting codes (Menorca
- Gianni, Mora
- 1987
(Show Context)
Citation Context ...s invertible modulo Qu, so the parametrization introduced in the above proposition makes sense. Inverting Q ′ u modulo Qu, we can write the alternative parametrization, reminiscent of the Shape Lemma =-=[24]-=-: Qu(u) = 0, ⎧ ⎪⎨ ⎪⎩ x1 = W1(u), . xn = Wn(u). Both forms of parametrizations will be used in the sequel, so we give them specific names. Formulae similar to those given in Proposition 2 can be found ... |

44 | Matrices in elimination theory
- Emiris, Mourrain
- 1999
(Show Context)
Citation Context ...er bases [9, 21], possibly followed by a Rational Univariate Representation [50]. We also mention the linear algebra methods, using the matrices introduced by Macaulay [44] or generalizations thereof =-=[19, 47, 18]-=-. 6sThe complexity of these zero-dimensional solving methods is not always known in terms of operations in the base field, which is here the rational function field k(P1, . . . , Pm). Moreover, there ... |

41 |
Schnelle Multiplikation großer Zahlen, Computing 7
- Schönhage, Strassen
- 1971
(Show Context)
Citation Context ...e multiplication of univariate polynomials of degree D, in terms of operations in the base ring. Mu(D) can be taken in O(D log D log log D), or Olog(D), using the algorithms of Schönhage and Strassen =-=[56]-=- and Schönhage [55]. • Ms(D, m) denotes the cost of m-variate series multiplication at precision D. This can be taken less than Mu((2D + 1) m ) using Kronecker’s substitution, see [39] and [64, ex. 16... |

40 |
On the complexity of semi-algebraic sets
- Heintz, Roy, et al.
- 1989
(Show Context)
Citation Context ...arametric system, we may consider systems with infinitesimal coefficients, seeing the infinitesimals as parameters. Solving such systems is a cornerstone of many algorithms in real algebraic geometry =-=[36, 37, 51]-=-. These algorithms often require to study the limits of the solutions when the infinitesimals go to zero, for which a parametric resolution is well-suited. As an example, we will show how to compute a... |

34 | Randomized matrix computations - Pan, Qian, et al. - 2012 |

31 |
Quadratic Newton iteration for systems with multiplicity
- Lecerf
(Show Context)
Citation Context ... ). This is most important for a naive multiplication algorithm where Mu(D) = O(D 2 ), or Karatsuba’s method, for which Mu(D) = O(D 1.59 ). The cost of the recombination of all factors is analyzed in =-=[41]-=-, and does not modify our complexity bound. Computing outside a given hypersurface. Suppose that we want the points V ′ ⊂ V lying outside a given hypersurface H ⊂ A m+n (k). It suffices to remove the ... |

31 | Efficient computation of minimal polynomials in algebraic extensions of finite fields
- Shoup
- 1999
(Show Context)
Citation Context ... a parametric resolution, computing this minimal polynomial is easy: a straightforward way to do so is the computation of the squarefree part of a resultant; a more efficient solution is described in =-=[61]-=-. This system is denoted Bershenko is the sequel. Our output was checked in a direct manner: since the relation R must be irreducible, it suffices to evaluate the relation R on the functions X/D, Y/D,... |

26 | Probabilistic algorithms for veri of polynomial identities - Schwartz - 1980 |

25 |
Deformation techniques for efficient polynomial equation solving
- Heintz, Krick, et al.
(Show Context)
Citation Context ...a parametric resolution have degree bounded by an intrinsic geometric quantity, which itself is bounded by the Bézout number of the input system. This result is the continuity of [53, 27] and notably =-=[34]-=-, and improves on the following important aspect. The results obtained in the above references require the zero-set of the defining system f = (f1, . . . , fn) to be in Noether position with respect t... |

25 | Sharp estimates for the arithmetic Nullstellensatz
- Krick, Pardo, et al.
- 2001
(Show Context)
Citation Context ..., and in particular to quantify the new possibilities of degeneracy. This requires to use the arithmetic forms of the geometric results used here; the kind of results we need is given for instance in =-=[38]-=-; the subsequent algorithm is given in the author’s PhD. Thesis [58]. • Our algorithm works only for the components were the Jacobian determinant is generically invertible. Recently, G. Lecerf propose... |

23 |
Diverses questions relatives au calcul formel avec des nombres algébriques
- Duval
- 1987
(Show Context)
Citation Context ... to the question of parametric systems is to give an exhaustive description of the solutions, describing all possibilities of degeneracy. We mention in particular the techniques of dynamic evaluation =-=[16, 29, 14]-=-, the comprehensive Gröbner bases [65] and the computation of parametric Gröbner bases proposed in [32] and [45]. Whereas the complexities of the dynamic evaluation method or of Montes’ algorithm are ... |

23 | On the time-space complexity of geometric elimination procedures
- Heintz, Matera, et al.
(Show Context)
Citation Context ...nally, we demonstrate the use of these results by treating various real-life applications. Related work. We have already mentioned that this article is in the continuity of the work of the TERA group =-=[27, 26, 25, 28, 35]-=- and notably of [34]. Let us mention other possible approaches. • Zero-dimensional solving over a rational function field. The resolution of the system as a zero-dimensional problem over the rational ... |

23 | A new algorithm for discussing gröbner bases with parameters
- Montes
(Show Context)
Citation Context ...es of degeneracy. We mention in particular the techniques of dynamic evaluation [16, 29, 14], the comprehensive Gröbner bases [65] and the computation of parametric Gröbner bases proposed in [32] and =-=[45]-=-. Whereas the complexities of the dynamic evaluation method or of Montes’ algorithm are not known to us, the approaches of Grigoriev and Vorobjov and of Weispfenning are known to lead to algorithms of... |

22 | Finding at least one point in each connected component of a re al algebraic set defined by a single equation
- Rouillier, Roy, et al.
- 2000
(Show Context)
Citation Context ...arametric system, we may consider systems with infinitesimal coefficients, seeing the infinitesimals as parameters. Solving such systems is a cornerstone of many algorithms in real algebraic geometry =-=[36, 37, 51]-=-. These algorithms often require to study the limits of the solutions when the infinitesimals go to zero, for which a parametric resolution is well-suited. As an example, we will show how to compute a... |

22 |
Bounds for traces in complete intersections and degrees in the Nullstellensatz
- Sabia, Solernó
- 1995
(Show Context)
Citation Context ...ients that appear in a parametric resolution have degree bounded by an intrinsic geometric quantity, which itself is bounded by the Bézout number of the input system. This result is the continuity of =-=[53, 27]-=- and notably [34], and improves on the following important aspect. The results obtained in the above references require the zero-set of the defining system f = (f1, . . . , fn) to be in Noether positi... |

20 |
The differential ideal [P
- Morrison
- 1999
(Show Context)
Citation Context ...h zero-set V(Jℓ). Then the Jacobian criterion given in [17, Corollary 16.16] states that each of the field extensions k(P1, . . . , Pm) → fr � � k[P1, . . . , Pm, X1, . . . , Xn]/Jℓ is separable (see =-=[46]-=- for a similar statement). Thus the first assertion of the lemma is proved; we now show that the degree of the extension K → B is indeed deg π. Using the separability condition obtained above, Proposi... |

19 |
Monomial bases and polynomial system solving
- Emiris, Rege
- 1994
(Show Context)
Citation Context ...er bases [9, 21], possibly followed by a Rational Univariate Representation [50]. We also mention the linear algebra methods, using the matrices introduced by Macaulay [44] or generalizations thereof =-=[19, 47, 18]-=-. 6sThe complexity of these zero-dimensional solving methods is not always known in terms of operations in the base field, which is here the rational function field k(P1, . . . , Pm). Moreover, there ... |

18 |
On the theoretical and practical complexity of the existential theory of the reals
- Heintz, Roy, et al.
- 1993
(Show Context)
Citation Context ...arametric system, we may consider systems with infinitesimal coefficients, seeing the infinitesimals as parameters. Solving such systems is a cornerstone of many algorithms in real algebraic geometry =-=[36, 37, 51]-=-. These algorithms often require to study the limits of the solutions when the infinitesimals go to zero, for which a parametric resolution is well-suited. As an example, we will show how to compute a... |

17 |
Sur la résolution des systèmes polynomiaux à paramètres
- Schost
- 2000
(Show Context)
Citation Context ...ilure, whose analysis requires to use arithmetic versions of Bézout’s theorem and of the Nullstellensatz. This is beyond the scope of this paper; such results may be found in the author’s PhD. Thesis =-=[58]-=-. Factorization. Suppose that the minimal polynomial of the specialized system splits into i irreducible factors of degrees (deg (1) π , . . . , deg (i) π ), so that this fiber can be described by i g... |

17 | Elliptic curves over and the computation of square roots modulo p - Schoof - 1985 |

16 |
Sur les variations séculaires des éléments elliptiques des sept planètes principales
- Verrier
(Show Context)
Citation Context ...s algorithm [6], evaluating the matrix Jac(Fκ) and the vector Fκ takes O(nL) operations in the quotient Hκ. All linear algebra takes O(n 4 ) operations in Hκ, using for instance Leverrier’s algorithm =-=[43]-=- for matrix inversion over a ring. All this sums up to O � (nL + n 4 )Mu(deg π)Ms(2 κ+1 , m) � operations in k, which proves the proposition. � 25s4.3 Recovering the coefficients Referring to the algo... |

15 |
Extension of the Berlekamp-Massey algorithm to N dimensions
- Sakata
- 1990
(Show Context)
Citation Context ...on is solved using Padé approximants, see [64]. In our general multivariate case, the question can be solved using linear algebra; other solutions based on Gröbner bases computations are presented in =-=[54, 22]-=-. We propose an algorithm with better complexity, which reduces to the usual computation of Padé approximants when m = 1. The algorithm is probabilistic: it requires to choose m − 1 values in the base... |

14 | Fast multivariate power series multiplication in characteristic zero
- Lecerf, Schost
- 2003
(Show Context)
Citation Context ...tution, see [39] and [64, ex. 16.16]. � �� ��� D+m If the base field k has characteristic zero, this complexity is in Olog Mu , i.e. m linear in the size of the series, up to logarithmic factors; see =-=[42]-=-. We make the assumption that there exists a universal constant c < 1 such that Ms(D, m) ≤ cMs(2D, m) holds for all D and m. With these notations, our first result concerns the existence and the compl... |

13 |
Triangularisation des systèmes constructibles - Applications à l’évaluation dynamique
- Dellière
- 1999
(Show Context)
Citation Context ... to the question of parametric systems is to give an exhaustive description of the solutions, describing all possibilities of degeneracy. We mention in particular the techniques of dynamic evaluation =-=[16, 29, 14]-=-, the comprehensive Gröbner bases [65] and the computation of parametric Gröbner bases proposed in [32] and [45]. Whereas the complexities of the dynamic evaluation method or of Montes’ algorithm are ... |

12 | Representation for the radical of a generated dierential ideal - Boulier, Lazard, et al. - 1995 |