Polynomials with integer roots

TL;DR

This paper investigates polynomials with integer roots, demonstrating that fixing any two consecutive coefficients results in only finitely many such polynomials for each degree.

Contribution

It establishes a finiteness result for polynomials with integer roots based on fixing two consecutive coefficients.

Findings

Finitely many polynomials for fixed consecutive coefficients

Results apply to polynomials of degree n ≥ 2

Provides structural insights into integer-root polynomials

Abstract

Let $\mathcal{F}_n$ be the set of unitary polynomials of degree $n \ge 2$ that have their roots in $\mathbb{Z}^*$. We note $$ Q(x) := x^n+a_{1}x^{n-1}+\dots+a_{n}. $$ We show that any two fixed consecutive coefficients $(a_{j},a_{j+1})$ ($j \in \{1,\dots,n-1\})$ define finitely many polynomials of $\mathcal{F}_n$.