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

2025/08/05 by Xiao, Stanley Yao
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.03848

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 ℚ is diagonalizable over ℚ.

Citations

• Exponential sums over singular binary quartic forms and applications
• Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves
• The density of discriminants of quintic rings and fields

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