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Fröberg's Conjecture for Quintics and Septics in Four Variables

Qihang Wang, Dongming Zhang

Over any characteristic-zero field, Wang and Zhang prove Fröberg’s predicted generic Hilbert series for equal-degree forms in four variables when the degree is 5 or 7 and the generator count is any r≥1. Exact rank certificates cover the previously open middle ranges r=6–11 and r=6–21, respectively; propagation covers the remaining counts.

Sources for this record

What was already known

The strong-Lefschetz cases covered r≤5. Boij, Dannetun and Lundqvist covered quintics at r≥12 and septics at r≥22; the middle ranges remained open.

AI involvement AI-centered

Wang and Zhang credit a GPT-5.6 Sol, Claude Fable 5 and Grok 4.6 workflow with mathematical ideas, proof strategies and exact certificates for the main results. They assembled the final argument.

The first-person disclosure connects model-origin mathematical ideas, strategies and exact certificates to the paper's main results. It does not establish AI leadership of the entire research project or assign each retained step.

Importance Notable

Completes the equal-degree quintic and septic cases in four variables.

Limitations

Other degrees, mixed degrees and positive characteristic remain outside the theorem. Programs certify the endpoint ranks exactly, but a completed check of the full propagation argument is not documented. Individual model contributions to the endpoint method are not identified.

Technical verification Partial

Standard-library programs reconstruct the ten quintic and fifteen septic endpoint matrices and certify maximal minors modulo 2, giving exact integer nonvanishing and characteristic-zero endpoint ranks. The paper gives the propagation and openness argument, but a completed human or machine check of that full mathematical reduction is not documented.

Formal checking

Reconstruct multiplication matrices and verify recorded nonsingular maximal minors modulo 2

Checker
The paper's Python verification programs, as reported by Wang and Zhang
Scope
Ten quintic and fifteen septic maximal endpoint ranks used in Theorem 1.1
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Few

The paper explains the reduction, but AI also helped organize and draft it; no named person's independent explanation of the central endpoint method is documented. Few is the evidence default, not a claim of incomprehension.

Correction and publication history