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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.24330 |
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| _version_ | 1866910072419909632 |
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| author | Swaminathan, Ashvin |
| author_facet | Swaminathan, Ashvin |
| contents | We determine the squarefree part of the scalar factor that arises when the quartic invariant of the generic binary form $F$ of odd degree $2n+1$ is expressed as the discriminant of the unique quadratic covariant $(F,F)_{2n}$. This squarefree part is exactly $p$ when $n+2$ is a power of an odd prime $p$, and $1$ otherwise. Equivalently, for each prime $p$: $v_2(S(n))$ is always even, and for odd $p$, $v_p(S(n))$ is odd if and only if $n+2$ is a power of $p$. This generalizes the classical identity $\operatorname{disc}(H(F))=-3\cdot\operatorname{disc}(F)$ for binary cubics, which dates back to the work of Cayley and Sylvester in the 1850s. The proof, which involves substantial explicit coefficient analysis and $p$-adic deformation arguments, was developed using an AI-assisted research workflow: the author's earlier partial attempts were completed through systematic collaboration with Claude Code (Anthropic) and Codex (OpenAI), and key arithmetic lemmas were formally verified in Lean~4 using Aristotle (Harmonic). We describe this workflow in detail as a case study in AI-assisted mathematical research. We also discuss representation-theoretic, geometric, and arithmetic interpretations of the quadratic covariant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24330 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Quartic Invariant of Odd Degree Binary Forms Swaminathan, Ashvin Number Theory 13A50, 11C08, 11B65, 14L24 We determine the squarefree part of the scalar factor that arises when the quartic invariant of the generic binary form $F$ of odd degree $2n+1$ is expressed as the discriminant of the unique quadratic covariant $(F,F)_{2n}$. This squarefree part is exactly $p$ when $n+2$ is a power of an odd prime $p$, and $1$ otherwise. Equivalently, for each prime $p$: $v_2(S(n))$ is always even, and for odd $p$, $v_p(S(n))$ is odd if and only if $n+2$ is a power of $p$. This generalizes the classical identity $\operatorname{disc}(H(F))=-3\cdot\operatorname{disc}(F)$ for binary cubics, which dates back to the work of Cayley and Sylvester in the 1850s. The proof, which involves substantial explicit coefficient analysis and $p$-adic deformation arguments, was developed using an AI-assisted research workflow: the author's earlier partial attempts were completed through systematic collaboration with Claude Code (Anthropic) and Codex (OpenAI), and key arithmetic lemmas were formally verified in Lean~4 using Aristotle (Harmonic). We describe this workflow in detail as a case study in AI-assisted mathematical research. We also discuss representation-theoretic, geometric, and arithmetic interpretations of the quadratic covariant. |
| title | On the Quartic Invariant of Odd Degree Binary Forms |
| topic | Number Theory 13A50, 11C08, 11B65, 14L24 |
| url | https://arxiv.org/abs/2603.24330 |