Computing Multiplicative Relations between Roots of a Polynomial

TL;DR

This paper introduces a new sufficient condition to identify polynomials with only trivial multiplicative relations among roots, and develops an efficient algorithm to determine these relations, enabling analysis of higher-degree polynomials previously intractable.

Contribution

The paper generalizes existing conditions for trivial multiplicative relations and provides an algorithm to decide and compute relations for polynomials in a new set E.

Findings

The algorithm efficiently handles high-degree polynomials.

Most polynomials are outside the set E, which is very small.

Numerical experiments confirm the algorithm's effectiveness.

Abstract

Multiplicative relations between the roots of a polynomial in $\mathbb{Q}[x]$ have drawn much attention in the field of arithmetic and algebra, while the problem of computing these relations is interesting to researchers in many other fields. In this paper, a sufficient condition is given for a polynomial $f\in\mathbb{Q}[x]$ to have only trivial multiplicative relations between its roots, which is a generalization of those sufficient conditions proposed in [C. J. Smyth, \emph{J. Number Theory}, 23 (1986), pp. 243--254], [G. Baron \emph{et al}., \emph{J. Algebra}, 177 (1995), pp. 827--846] and [J. D. Dixon, \emph{Acta Arith.} 82 (1997), pp. 293--302]. Based on the new condition, a subset $E\subset\mathbb{Q}[x]$ is defined and proved to be genetic (i.e., the set $\mathbb{Q}[x]\backslash E$ is very small). We develop an algorithm deciding whether a given polynomial $f\in\mathbb{Q}[x]$ is in $E$ and returning a basis of the lattice consisting of the multiplicative relations between the roots of $f$ whenever $f\in E$. The numerical experiments show that the new algorithm is very efficient for the polynomials in $E$. A large number of polynomials with much higher degrees, which were intractable before, can be handled successfully with the algorithm.