Covariants and simultaneous diagonalization of pairs of ternary quadratic forms, and binary quartic forms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915429985812480 |
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| author | Xiao, Stanley Yao |
| author_facet | Xiao, Stanley Yao |
| contents | 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 $\mathbb{Q}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_03848 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Covariants and simultaneous diagonalization of pairs of ternary quadratic forms, and binary quartic forms Xiao, Stanley Yao Number Theory 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 $\mathbb{Q}$. |
| title | Covariants and simultaneous diagonalization of pairs of ternary quadratic forms, and binary quartic forms |
| topic | Number Theory |
| url | https://arxiv.org/abs/2508.03848 |