Fröberg's Conjecture for Quintics and Septics in Four Variables

Let k be a field of characteristic zero and let S = k[x_1,x_2,x_3,x_4]. We prove Fröberg's predicted Hilbert series for ideals generated by r general forms of equal degree d for every r ≥ 1 in each of the two cases d = 5 and d = 7. Relative to the classical cases r ≤ 5 and the equal-degree theorem through degree d + 2 of Boij–Dannetun–Lundqvist, the generator-count ranges requiring new input are 6 ≤ r ≤ 11 for quintics and 6 ≤ r ≤ 21 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. Zariski openness then gives the result over every characteristic-zero field. The unrestricted Fröberg conjecture remains outside the scope of the paper.