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On the Quartic Invariant of Odd Degree Binary Forms

Ashvin Swaminathan

Abstract

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.

On the Quartic Invariant of Odd Degree Binary Forms

Abstract

We determine the squarefree part of the scalar factor that arises when the quartic invariant of the generic binary form of odd degree is expressed as the discriminant of the unique quadratic covariant . This squarefree part is exactly when is a power of an odd prime , and otherwise. Equivalently, for each prime : is always even, and for odd , is odd if and only if is a power of . This generalizes the classical identity 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 -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.

Paper Structure

This paper contains 28 sections, 26 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. The classical case of binary cubics
  3. The generalization to odd degree
  4. Main result
  5. Outline of the proof
  6. History and methodology
  7. Preliminaries and normalization
  8. Representation-theoretic uniqueness
  9. Discriminant expansion and explicit coefficient extraction
  10. Test monomials
  11. Closed formula for $C_{n,r}$
  12. Factoring $C_{n,r}$
  13. From explicit coefficients to the content
  14. The binomial gcd theorem
  15. Relating the gcd to $S(n)$
  16. ...and 13 more sections

Key Result

Theorem 1.1

For every integer $n \ge 1$ and every prime $p$, the parity of $v_p(S(n))$ is determined as follows: Equivalently, where $\mathop{\mathrm{sqf}}\nolimits$ denotes the squarefree part.

Theorems & Definitions (65)

  • Theorem 1.1
  • Example 1.2
  • Remark
  • Remark
  • Definition 2.1: Transvectant
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Definition 2.4
  • ...and 55 more