Binary forms with the same value set I

TL;DR

This paper develops a theoretical framework to determine when two binary forms of degree at least three have identical value sets, focusing on forms with small automorphism groups and reducing the problem to lattice covering questions.

Contribution

It introduces a general theory linking value set equality to automorphism groups and lattice coverings, specifically addressing forms with small automorphism groups.

Findings

Reduces the problem to lattice covering questions in f^2.

Provides solutions for forms with small automorphism groups.

Lays groundwork for analyzing forms with larger automorphism groups in subsequent parts.

Abstract

Given a binary form $F \in \mathbb{Z}[X, Y]$, we define its value set to be $\{F(x, y) : (x, y) \in \mathbb{Z}^2\}$. Let $F, G \in \mathbb{Z}[X, Y]$ be two binary forms of degree $d \geq 3$ and with non-zero discriminant. In a series of three papers, we will give necessary and sufficient conditions on $F$ and $G$ to have the same value set. These conditions will be entirely in terms of the automorphism groups of the forms. In this paper, we will build the general theory that reduces the problem to a question about lattice coverings of $\mathbb{Z}^2$, and we solve this problem when $F$ and $G$ have a small automorphism group. The larger automorphism groups $D_4$ and $D_3, D_6$ will respectively be treated in part II and part III.