On the irreducibility of the non-cyclotomic part of most 0,1-polynomials with few terms

TL;DR

This paper demonstrates that for most choices of exponents, the non-cyclotomic part of sparse 0,1-polynomials is irreducible, extending Schinzel's result with an alternative proof approach.

Contribution

It provides an alternative exposition of Schinzel's result showing irreducibility of the non-cyclotomic part of sparse polynomials for almost all exponent choices.

Findings

Most such polynomials are irreducible after removing cyclotomic factors.

The result holds for almost all increasing sequences of positive integers.

The paper offers a new proof approach to Schinzel's theorem.

Abstract

We provide an alternative exposition of a result due to Schinzel. Fix an integer $k \ge 2$. For almost all choices of positive integers $n_{1} < \cdots < n_{k}$, we show that the polynomial $F(x) = 1 + x^{n_{1}} + \cdots + x^{n_{k}}$, removed of its cyclotomic factors, is irreducible.