Counting irreducible polynomials with prescribed coefficients over a finite field

Gao, Zhicheng; Kuttner, Simon; Wang, Qiang

doi:10.1016/j.ffa.2022.102023

Cited by 8 publications

(6 citation statements)

References 21 publications

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“…In this section, we use the generating function method developed in [11] to study the distribution of Y k . Given ε ∈ E, we shall also use M d (ε) to denote the set of polynomials in M d which are equivalent to ε.…”

Section: Generating Functions Probabilities and Momentsmentioning

confidence: 99%

Asymptotic distribution of the number of zeros in random polynomials in a given equivalence class over a finite field

Gao¹

2022

Preprint

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Hayes equivalence is defined on monic polynomials over a finite field Fq in terms of the prescribed leading coefficients and the residue classes modulo a given monic polynomial Q. We study the distribution of the number of zeros in a random polynomial over finite fields in a given Hayes' equivalence class. It is well known that the number of distinct zeros of a random polynomial over Fq is asymptotically Poisson with mean 1. We show that this is also true for polynomials in any given Hayes equivalence class provided that the degree of freedom goes to infinity. Asymptotic formulas are also given for the number of such polynomials when the degree of Q and the number of prescribed leading coefficients are small compared with the degree of the polynomial. When the equivalence class is defined by leading coefficients only, the problem is equivalent to the study of the distance distribution over Reed-Solomon codes and our asymptotic formulas extend some earlier results.

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Section: Generating Functions Probabilities and Momentsmentioning

confidence: 99%

Asymptotic distribution of the number of zeros in random polynomials in a given equivalence class over a finite field

Gao¹

2022

Preprint

Self Cite

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“…In this section, we introduce the necessary background to be able to count monic polynomials with the first w prescribed coefficients using the generating functions method. A general combinatorial framework for counting irreducible polynomials with prescribed coefficients, using generating functions with coefficients from a group algebra, was developed in [5] and Section 2 of [6].…”

Section: Definitions and Notationsmentioning

confidence: 99%

“…Proposition 1. (Proposition 1, [6]) G is an abelian group under multiplication f g = f g with identity 1 .…”

Section: Definitions and Notationsmentioning

confidence: 99%

“…In particular, we re-derive the total number of irreducible polynomials, and the number of irreducible polynomials of the form x m + βx m−1 + g(x) where β ∈ F q is fixed, and g(x) ∈ F q [x] of degree at most m − 2 is varied. More details can be found in [5,6], where this method was first introduced, and different cases such as prescribed trace and norm, or prescribed multiple coefficients were considered respectively.…”

Section: Counting Irreducible Polynomialsmentioning

confidence: 99%

“…In Section 2 we provide background definitions and preliminary results that were first introduced in [5,6], such as generating functions defined on the group algebra of the equivalent classes for polynomials over finite fields. In Section 3 we demonstrate the generating function method over group rings to count irreducible polynomials with prescribed coefficients, which were explored in [5,6] earlier. In Section 4 we develop the general results on counting polynomials with prescribed coefficients and prescribed factorization pattern.…”

Section: Introductionmentioning

confidence: 99%

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