On the Complexity Exponent of Polynomial System Solving

Hoeven, Joris van der; Lecerf, Grégoire

doi:10.1007/s10208-020-09453-0

Cited by 17 publications

(12 citation statements)

References 64 publications

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“…More precisely, we make use of the recent work of Safey El Din and Schost [61], who take into account multi-homogeneity and provide estimates on the height of the representations and the bit complexities of their algorithms. Note that as this work reached completion, a new preprint by van der Hoeven and Lecerf [66] appeared that points to the possibility of improving further the exponent of d n in our results, while retaining the same approach. The Kronecker representation, and similar constructions, have also appeared in the literature under the name 'rational univariate representation' [60,5].…”

Section: Previous Workmentioning

confidence: 66%

Effective coefficient asymptotics of multivariate rational functions via semi-numerical algorithms for polynomial systems

Melczer

Salvy

2021

Journal of Symbolic Computation

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The coefficient sequences of multivariate rational functions appear in many areas of combinatorics. Their diagonal coefficient sequences enjoy nice arithmetic and asymptotic properties, and the field of analytic combinatorics in several variables (ACSV) makes it possible to compute asymptotic expansions. We consider these methods from the point of view of effectivity. In particular, given a rational function, ACSV requires one to determine a (generically) finite collection of points that are called critical and minimal. Criticality is an algebraic condition, meaning it is well treated by classical methods in computer algebra, while minimality is a semi-algebraic condition describing points on the boundary of the domain of convergence of a multivariate power series. We show how to obtain dominant asymptotics for the diagonal coefficient sequence of multivariate rational functions under some genericity assumptions using symbolic-numeric techniques. To our knowledge, this is the first completely automatic treatment and complexity analysis for the asymptotic enumeration of rational functions in an arbitrary number of variables.We also propose an algorithm finding minimal critical points in many cases, even when combinatoriality is not assumed, at the price of an increase in complexity. Our result in that case is the following, which is stated precisely in Theorem 57 below.Result 2. Let F (z) ∈ Z(z 1 , . . . , z n ) be a rational function with numerator and denominator of degrees at most d and coefficients of absolute value at most 2 h . Assuming that F satisfies certain verifiable assumptions stated in Section 3.3, then F admits a finite number of minimal critical points that can be determined iñ O hd 9n+5 2 3n bit operations. From there, the asymptotics of the diagonal coefficients follow with the same complexity as in Result 1.Aside from the existence of minimal critical points, we conjecture that the assumptions on F required to apply Theorem 57 in the non-combinatorial case hold generically. dimensional.We start with the extended system, the other one is similar. This system has n + 2 equations of degree at most d in n+2 variables. Example 36 shows that it is evaluated by a straight-line program of length O(L+n). We partition the variables into three blocks z, λ, t.The bound C n (d) is obtained as the sum of the non-zero coefficients of θ z , θ λ , θ t inleading to C n (d) = nd n+1 . The computation for the height is similar. With η 1 = h + log(n + 1)d, η 2 = · · · = η n+1 = h + d + log(n + 1)d + 1, η n+2 = h + 2 log(n + 1)dandit follows that H n (η, d) = O(Dn(d + n)(h + log(n + 1)d)) =Õ(dD(h + d)). The complexity then follows from injecting these quantities in the previous proposition. The bounds for the system formed by the critical point equations only are derived as above with simpler computationsleading to nD for the degree andÕ (D(h + d)) for the height.Example 44. These inequalities reflect a perceptible growth of the sizes with the number of variables in computations.Starting from the same system as in Ex...

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Section: Previous Workmentioning

confidence: 66%

Effective coefficient asymptotics of multivariate rational functions via semi-numerical algorithms for polynomial systems

Melczer

Salvy

2021

Journal of Symbolic Computation

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“…An important application of the present results concerns polynomial system solving, for which we prove new complexity bounds in [28]: the key algorithms are the Kronecker solver [14] and fast multivariate modular composition. For the latter problem, we mostly follow the strategy deployed in the proof of Proposition 5.7.…”

Section: Discussionmentioning

confidence: 88%

“…The new complexity bounds for multi-point evaluation are also crucial for our new bit complexity bounds for multivariate modular composition and the application to polynomial system solving in [28]. Section 7 addresses the special case when is a field of small positive characteristic p. We closely revisit the method proposed in [34, section 6], and again make the complexity bound more explicit.…”

Section: Contributionsmentioning

confidence: 99%

Fast multivariate multi-point evaluation revisited

Hoeven

Lecerf

2020

Journal of Complexity

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In 2008, Kedlaya and Umans designed the first multivariate multi-point evaluation algorithm over finite fields with an asymptotic complexity that can be made arbitrarily close to linear. However, it remains a major challenge to make their algorithm efficient for practical input sizes. In this paper, we revisit and improve their algorithm, while keeping this ultimate goal in mind. In addition we sharpen the known complexity bounds for modular composition of univariate polynomials over finite fields.N mod p i are handled recursively, for i = 1, …, s. Modular compositionLet us first discuss the standard modular composition problem when n= 1. Let f , g and h be polynomials in [x] of respective degrees

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“…This problem naturally relates to several areas of applied algebra, including polynomial system solving, since we may use it to verify whether all points in a given set are solutions to a system of polynomial equations. In [16], it has even be shown that efficient algorithms for multi-point evaluation lead to efficient algorithms for polynomial system solving. As an other more specific application, bivariate polynomial evaluation intervenes in computing generator matrices of geometric error correcting codes [20].…”

Section: Introductionmentioning

confidence: 99%

Fast amortized multi-point evaluation

Hoeven

Lecerf

2021

Journal of Complexity

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The efficient evaluation of multivariate polynomials at many points is an important operation for polynomial system solving. Kedlaya and Umans have recently devised a theoretically efficient algorithm for this task when the coefficients are integers or when they lie in a finite field. In this paper, we assume that the set of points where we need to evaluate is fixed and "sufficiently generic". Under these restrictions, we present a quasi-optimal algorithm for multi-point evaluation over general fields. We also present a quasi-optimal algorithm for the opposite interpolation task. * . This paper is part of a project that has received funding from the French "Agence de l'innovation de défense".

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On the Complexity Exponent of Polynomial System Solving

Cited by 17 publications

References 64 publications

Effective coefficient asymptotics of multivariate rational functions via semi-numerical algorithms for polynomial systems

Effective coefficient asymptotics of multivariate rational functions via semi-numerical algorithms for polynomial systems

Fast multivariate multi-point evaluation revisited

Fast amortized multi-point evaluation

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