Discoveries / algebra

Fröberg’s conjecture for quintics and septics in four variables

Let $S=k[x_1,x_2,x_3,x_4]$ over any characteristic-zero field. For each $d\in\{5,7\}$ and every $r\ge1$, the paper proves that $r$ general degree-$d$ forms satisfy Fröberg’s predicted Hilbert series $$ \operatorname{HS}_{S/(F_1,\ldots,F_r)}(t) = \left[\frac{(1-t^d)^r}{(1-t)^4}\right]_+. $$ The genuinely new ranges are $6\le r\le11$ for quintics and $6\le r\le21$ for septics. These are reduced to finitely many exact rank tests of Macaulay multiplication matrices; explicit maximal minors are nonzero mod $2$, so the required ranks hold in characteristic zero. Thus the conjecture is completely settled for the equal-degree four-variable slices $d=5$ and $d=7$, but not in general.