Fr\"oberg's Conjecture for Quintics and Septics in Four Variables

Abstract

Let $k$ be a field of characteristic zero and let $S=k[x_1,x_2,x_3,x_4]$. We prove Fr\"oberg's predicted Hilbert series for ideals generated by $r$ general forms of equal degree $d$ for every $r\geq1$ in each of the two cases $d=5$ and $d=7$. Relative to the classical cases $r\leq5$ and the equal-degree theorem through degree $d+2$ of Boij--Dannetun--Lundqvist, the generator-count ranges requiring new input are $6\leq r\leq11$ for quintics and $6\leq r\leq21$ for septics. The proof reduces each slice to finitely many endpoint ranks of Macaulay multiplication matrices. For quintics, ten exact endpoint computations based on twenty-one sparse forms suffice. For septics, a nested family of 120 integral forms supplies fifteen endpoint computations. In every endpoint certificate for these new ranges, an explicitly recorded maximal minor is nonzero modulo $2$, hence is a nonzero integer. The case $r=5$ is the classical strong Lefschetz instance; for quintics we also record a matching modular rank and Koszul bound. Zariski openness then gives the result over every characteristic-zero field. The unrestricted Fr\"oberg conjecture remains outside the scope of the paper. The main results of this paper were obtained through a generative-AI workflow using OpenAI GPT-5.6 Sol, Anthropic Claude Fable 5, and Grok 4.6. Further details appear in the disclosure at the end of the paper.