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Irreducibility of lacunary polynomials with 0,1 coefficients

Alexandros Kalogirou

Abstract

We show that $0,1$-polynomials of high degree and few terms are irreducible with high probability. Formally, let $k\in\mathbb{N}$ and $F(x)=1+\sum_{i=1}^kx^{n_i}$, where $ 0<n_1<\cdots<n_k\leq N. $ Then we show that $\lim_{k\rightarrow\infty}\limsup_{N\rightarrow\infty}\mathbb{P}(\text{$F(x)$ is reducible})=0.$ The probability in this context is derived from the uniform count of polynomials $F(x)$ of the above form.

Irreducibility of lacunary polynomials with 0,1 coefficients

Abstract

We show that -polynomials of high degree and few terms are irreducible with high probability. Formally, let and , where Then we show that F(x) The probability in this context is derived from the uniform count of polynomials of the above form.

Paper Structure

This paper contains 7 sections, 4 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries I
  3. Preliminaries II
  4. Divisibility by $\Phi_n(x)$, $n\leq k/\exp(\sqrt{\log k})$
  5. Divisibility by $\Phi_n(x)$ for $k/\exp(\sqrt{\log k})<n\leq e^{k/(24\log k)}$
  6. $e^{k/(24\log k)}\leq n\leq N\log N$
  7. Acknowlegments

Key Result

Theorem 1

For $F(x)$ as above, we have Here the probability is defined as the proportion of $F(x)$ as above that are reducible.

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Lemma 1
  • proof