Covariants and simultaneous diagonalization of pairs of ternary quadratic forms, and binary quartic forms

TL;DR

This paper establishes a correspondence between covariants of binary quartic forms and pairs of ternary quadratic forms, providing a canonical method for diagonalization of quadratic form pairs over fields of characteristic zero.

Contribution

It introduces a novel correspondence linking covariants of binary quartic forms with those of ternary quadratic forms, enabling canonical diagonalization over any characteristic zero field.

Findings

Canonical degree six covariant of binary quartic forms linked to a cubic covariant of ternary quadratic forms

A canonical diagonalization method for pairs of n-ary quadratic forms over any characteristic zero field

A criterion for diagonalizability of quadratic form pairs over $\

Abstract

In this paper we prove a correspondence between a canonical degree six covariant of binary quartic forms $F$ and a cubic covariant of a pair of ternary quadratic forms $(f_A, f_B)$. In the process we obtain a canonical way to diagonalize a pair of $n$-ary quadratic forms over any field $K$ of characteristic zero. As a corollary, we give a precise criterion to decide whether a pair of $n$-ary quadratic forms over $\mathbb{Q}$ is diagonalizable over $\mathbb{Q}$.