The quartic invariant of a binary form F of odd degree 2n+1 can be expressed as the discriminant of its unique quadratic covariant (F,F)_{2n}. But the relationship involves a scalar factor that depends on the degree. What is this factor?
The paper on the quartic invariant (arXiv: 2603.24330) determines its squarefree part: it equals the prime p when n+2 is a power of an odd prime p, and equals 1 otherwise. This generalizes a classical identity of Cayley and Sylvester for binary cubics (degree 3, where n=1 and the factor involves the prime 3).
The result links invariant theory (19th-century algebra) to prime factorization (number theory) through a condition on the degree: the arithmetic nature of n+2 determines the algebraic structure of the invariant.
Notably, the paper's methodology explicitly credits AI tools: Claude Code and Codex assisted in completing proofs, and formal verification was conducted in Lean 4. The combination of symbolic computation, AI-assisted reasoning, and formal verification produced a result in classical algebra that had remained open.
The through-claim: the squarefree part of an invariant-theory scalar is determined by the prime factorization of the degree. A question from 19th-century algebra — what scalar connects two classical invariants? — has an answer in 21st-century number theory, found with 21st-century tools. The primality of n+2 controls the algebraic relationship. The degree's arithmetic is the invariant's structure.
2603.24330. Algebra / invariant theory / binary forms / discriminants / AI-assisted proof.