Fröberg's Conjecture for Quintics and Septics in Four Variables

This deposit contains the preprint and source package for a proof of Fröberg’s predicted Hilbert series for ideals generated by general forms of equal degree in four variables over fields of characteristic zero. For each fixed generator count r ≥ 1, the result establishes the conjectured series separately for degrees 5 and 7 on a nonempty Zariski-open subset. Relative to the cited general results, the new ranges are 6 ≤ r ≤ 11 for quintics and 6 ≤ r ≤ 21 for septics. The proof reduces the problem to finitely many exact endpoint ranks of Macaulay multiplication matrices, certifies nonzero maximal minors modulo 2, and uses Zariski openness and endpoint propagation to obtain the characteristic-zero statements. The result does not resolve the unrestricted Fröberg conjecture, mixed degrees, other numbers of variables, or positive characteristic.