Binary forms with the same value set III. The case of ${\bf D}_3$ and ${\bf D_6}$

TL;DR

This paper characterizes when binary forms of degree at least 3 with automorphism groups isomorphic to dihedral groups of order 6 or 12 have identical value sets over the integers.

Contribution

It provides a complete characterization of when such binary forms share the same integer value set, focusing on forms with specific dihedral automorphism groups.

Findings

Identifies conditions for binary forms with dihedral automorphism groups to have identical value sets.

Establishes a precise criterion for equality of value sets in the case of ${f D}_3$ and ${f D}_6$ automorphism groups.

Contributes to the classification of binary forms based on their value sets and automorphism groups.

Abstract

Let $F, G \in \mathbb{Z}[X, Y]$ be binary forms of degree $\geq 3$ with automorphism groups isomorphic to the dihedral group of cardinality $6$ or $12$. We characterize exactly when $F$ and $G$ have the same value set, i.e. $F(\mathbb{Z}^2) = G(\mathbb{Z}^2)$.