## Hermite equivalence of polynomials

### Volume 209 / 2023

#### Abstract

We compare results of Birch and Merriman (1972), Győry (1973, 1974) and Evertse and Győry (1991, 2017), which imply in a precise form that polynomials in $\mathbb {Z} [X]$ of given discriminant lie in finitely many ${\rm GL}_2(\mathbb {Z} )$-equivalence classes, with a forgotten theorem of Hermite (1854, 1857) which implies that polynomials in $\mathbb {Z} [X]$ with given discriminant lie in finitely many equivalence classes with respect to a weaker notion of equivalence, henceforth called “Hermite equivalence”. We show that ${\rm GL}_2(\mathbb {Z})$-equivalent polynomials are Hermite equivalent and that for polynomials of degree $2$ or $3$ the converse is also true. On the other hand, for every $n\geq 4$ we give an infinite class 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. One of the constructions (see Theorem 5.1) uses an irreducibility result for a certain class of polynomials which may be of independent interest.

We point out that the results of Birch and Merriman, Győry, and Evertse and Győry are in general much more precise than Hermite’s theorem. As a consequence, we correct a faulty reference occurring in Narkiewicz’ excellent book (2019), where ${\rm GL}_2 (\mathbb {Z})$-equivalence and Hermite equivalence of polynomials were mixed up.