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Factorization type probabilities of polynomials with prescribed coefficients over a finite field

Kaloyan Slavov

Abstract

Let $f(T)$ be a monic polynomial of degree $d$ with coefficients in a finite field $\mathbb{F}_q$. Extending earlier results in the literature, but now allowing $(q,2d)>1$, we give a criterion for $f$ to satisfy the following property: for all but $d^2-d-1$ values of $s$ in $\mathbb{F}_q$, the probability that $f(T)+sT+b$ is irreducible over $\mathbb{F}_q$ (as $b\in\mathbb{F}_q$ is chosen uniformly at random) is $1/d+O(q^{-1/2})$.

Factorization type probabilities of polynomials with prescribed coefficients over a finite field

Abstract

Let be a monic polynomial of degree with coefficients in a finite field . Extending earlier results in the literature, but now allowing , we give a criterion for to satisfy the following property: for all but values of in , the probability that is irreducible over (as is chosen uniformly at random) is .

Paper Structure

This paper contains 2 sections, 6 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. The proof

Key Result

Theorem 1

Let $f(T)\in\mathbb{F}_q[T]$ be a monic polynomial of degree $d$. Suppose $(q,d(d-1))=1$. Then $f$ satisfies (*) with $m=1$.

Theorems & Definitions (14)

  • Theorem 1
  • Proposition 2
  • Remark 3
  • Theorem 4
  • Corollary 5
  • Example 6
  • Remark 7
  • Conjecture 8
  • Lemma 9
  • proof
  • ...and 4 more