Indecomposable polynomials and their spectrum

Bodin, Arnaud; Dèbes, Pierre; Najib, Salah

doi:10.4064/aa139-1-7

Cited by 19 publications

(17 citation statements)

References 10 publications

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“…We contribute to these approaches by implementing some ideas and results coming from connected areas, notably of Grothendieck (Arithmetic Geometry) and Gao (Polyhedral Combinatorics). This leads to new answers to the problem together with an improved and unified presentation of results from our previous papers [BDN09a] [BDN09b] and other related papers. Those were concerned with special cases of the general situation considered here.…”

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confidence: 89%

“…It is not as good as (4); the advantage of B F lies in its full explicitness (which may lead to better bounds in specific cases (see §2.2.6)) and in its arithmetic meaning, where the name "bad prime divisor" originates: if B F (t * ) = 0, the distinct roots (in k(t)) of ∆ F remain defined and distinct after specialization of t to t * ∈ k s . The construction improves on [BDN09a,§3], which used a result of Zannier rather than the Grothendieck reduction theory. We also explain how to get rid of the "monic" assumption in corollary 2.8, to relax the condition on the characteristic of k and to pass from 2 to any number ℓ of indeterminates.…”

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confidence: 99%

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Families of polynomials and their specializations

Bodin

Dèbes

Najib

2017

Journal of Number Theory

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Abstract. For a polynomial in several variables depending on some parameters, we discuss some results to the effect that for almost all values of the parameters the polynomial is irreducible. In particular we recast in this perspective some results of Grothendieck and of Gao.This paper is devoted to irreducibility questions for families of polynomials in several indeterminates x 1 , . . . , x ℓ parametrized by further indeterminates t 1 , . . . , t s . We assume that ℓ 2 and the base field k is algebraically closed; the more arithmetic case ℓ = 1 depends on the base field and involves different tools and techniques.Set(where k(t) is the algebraic closure of k(t)); F is said to be generically irreducible. The core question is about the irreducibility of the polynomials obtained by substituting elements t * 1 , . . . , t * s ∈ k for the corresponding parameters t 1 , . . . , t s -the specializations of F .More specifically we wish to investigate the following problem, as explicitly as possible: -when the generic irreducibility property is satisfied, show some boundedness results on the following set, which we call the spectrum of F :}, and some density results for its complement, -find some criteria for the generic irreduciblity property to be satisfied and deduce some new specific examples.A first approach rests on classical results of Noether and Bertini and a second one involves more combinatorial tools like the Newton polygon and the

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confidence: 89%

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confidence: 99%

Families of polynomials and their specializations

Bodin

Dèbes

Najib

2017

Journal of Number Theory

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“…, x ν ] for some value c ∈ F q then P is indecomposable. Indecomposable polynomials have a special interest in regards of Stein's theorem and provide families of irreducible polynomials (see [15], [12], [2]):…”

Section: Indecomposable Polynomialsmentioning

confidence: 99%

Generating series for irreducible polynomials over finite fields

Bodin

2010

Finite Fields and Their Applications

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“…[Wat08], for effective results on the reduction modulo a prime number of a non-composite polynomial or a rational function see e.g. [CN10,BDN09,BCN], for combinatorial results see e.g. [Gat08].…”

Section: Introductionmentioning

confidence: 99%

A recombination algorithm for the decomposition of multivariate rational functions

Chèze

2012

Math. Comp.

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Abstract. In this paper we show how we can compute in a deterministic way the decomposition of a multivariate rational function with a recombination strategy. The key point of our recombination strategy is the used of Darboux polynomials. We study the complexity of this strategy and we show that this method improves the previous ones. In appendix, we explain how the strategy proposed recently by J. Berthomieu and G. Lecerf for the sparse factorization can be used in the decomposition setting. Then we deduce a decomposition algorithm in the sparse bivariate case and we give its complexity.

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Indecomposable polynomials and their spectrum

Cited by 19 publications

References 10 publications

Families of polynomials and their specializations

Families of polynomials and their specializations

Generating series for irreducible polynomials over finite fields

A recombination algorithm for the decomposition of multivariate rational functions

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