Hermite equivalence of polynomials

TL;DR

This paper revisits Hermite's old notion of polynomial equivalence, revealing its connections to invariant theory and comparing it with modern equivalence concepts, with new results for degrees 2, 3, and higher.

Contribution

It provides a natural interpretation of Hermite equivalence via invariant rings and ideals, and clarifies its relationship with ${ m GL}_2(bZ)$-equivalence, including new examples for degrees ≥4.

Findings

Hermite equivalence relates to invariant rings and ideals.

${ m GL}_2(bZ)$-equivalence implies Hermite equivalence.

For degrees 2 and 3, the two notions coincide; for higher degrees, they differ.

Abstract

In this paper, we resurrect a long-forgotten notion of equivalence for univariate polynomials with integral coefficients introduced by Hermite in the 1850s. We show that the Hermite equivalence class of a polynomial has a very natural interpretation in terms of the invariant ring and invariant ideal associated with the polynomial. We apply this interpretation to shed light on the relationship between Hermite equivalence and more familiar notions of polynomial equivalence, such as ${\rm GL}_2(\mathbb{Z})$- and $\mathbb{Z}$-equivalence. Specifically, we prove that ${\rm GL}_2(\mathbb{Z})$-equivalent polynomials are Hermite equivalent and, for polynomials of degree $2$ or $3$, the converse is also true. On the other hand, for every $n\geq 4$, we give infinite collections of examples of polynomials $f,g\in \mathbb{Z}[X]$ of degree $n$ that are Hermite equivalent but not ${\rm GL}_2(\mathbb{Z})$-equivalent.