Binary forms with the same value set II. The case of ${\bf D}_4$

TL;DR

This paper proves that for binary forms of degree at least 3 with automorphism group isomorphic to D4, having the same integer value set implies they are equivalent under GL(2,Z) transformations.

Contribution

It establishes a uniqueness result for binary forms with automorphism group D4 based on their value sets, extending understanding of form equivalences.

Findings

Forms with the same value set are GL(2,Z)-equivalent

Automorphism group D4 plays a key role in form classification

Results apply to forms with degree ≥ 3 and non-zero discriminant

Abstract

Let $F, G \in \mathbb{Z}[X, Y]$ be binary forms of degree $\geq 3$, non-zero discriminant and with automorphism group isomorphic to $D_4$. If $F(\mathbb{Z}^2) = G(\mathbb{Z}^2)$, we show that $F$ and $G$ are ${\rm GL}(2, \mathbb{Z})$--equivalent.