Bounded-degree factors of lacunary multivariate polynomials

Grenet, Bruno

doi:10.1016/j.jsc.2015.11.013

Cited by 8 publications

(2 citation statements)

References 39 publications

(68 reference statements)

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“…Testing the irreducibility of lacunary polynomials or computing the greatest common divisor of two lacunary polynomials are NP-hard problems [5][6][7]. Computing the irreducible factors of bounded degree of lacunary polynomials can be accomplished in polynomial time [8] as well as computing the integer roots of lacunary integer polynomials [9].…”

Section: Introductionmentioning

confidence: 99%

Hard to Detect Factors of Univariate Integer Polynomials

Dennunzio,

Formenti,

Margara

2023

Mathematics

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We investigate the computational complexity of deciding whether a given univariate integer polynomial p(x) has a factor q(x) satisfying specific additional constraints. When the only constraint imposed on q(x) is to have a degree smaller than the degree of p(x) and greater than zero, the problem is equivalent to testing the irreducibility of p(x) and then it is solvable in polynomial time. We prove that deciding whether a given monic univariate integer polynomial has factors satisfying additional properties is NP-complete in the strong sense. In particular, given any constant value k∈Z, we prove that it is NP-complete in the strong sense to detect the existence of a factor that returns a prescribed value when evaluated at x=k (Theorem 1) or to detect the existence of a pair of factors—whose product is equal to the original polynomial—that return the same value when evaluated at x=k (Theorem 2). The list of all the properties we have investigated in this paper is reported at the end of Section 1.

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Section: Introductionmentioning

confidence: 99%

Hard to Detect Factors of Univariate Integer Polynomials

Dennunzio,

Formenti,

Margara

2023

Mathematics

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“…The factorization problem for sparse polynomials can be vaguely stated as the question of whether the irreducible factors of a sparse polynomial are also sparse, apart from obvious exceptions. Aspects of this problem have been studied in various settings and for different formalizations of the notion of sparsenness, see for instance [Len99, Sch00, KK06, AKS07, FGS08, Gre16,ASZ17]. Several of these studies were based on tools from Diophantine geometry like lower bounds for the height of points and subvarieties, and unlikely intersections of subvarieties and subgroups of a torus.…”

Section: Introductionmentioning

confidence: 99%

Factorization of bivariate sparse polynomials

Amoroso

Sombra

2019

Acta Arith.

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We prove a function field analogue of a conjecture of Schinzel on the factorization of univariate polynomials over the rationals. We derive from it a finiteness theorem for the irreducible factorizations of the bivariate Laurent polynomials in families with fixed set of complex coefficients and varying exponents. Roughly speaking, this result shows that the truly bivariate irreducible factors of these sparse Laurent polynomials, are also sparse. The proofs are based on a variant of the toric Bertini's theorem due to Zannier and Fuchs, Mantova and Zannier.Date: 12/9/2018.

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