Hilbert Series in Four Variables and Linear Factors of High Multiplicity

Abstract

Explicit sequences of forms with the Hilbert series predicted by Fröberg's conjecture are difficult to construct beyond the almost complete intersection case. Over a field of characteristic zero, we prove that six forms of degree $d\ge 4$ in four variables cannot have the predicted series if each is divisible by a power of a linear form with exponent at least $(15d+91)/16$. A power of the product of six quadric cones gives a nonzero element of the inverse system in a degree where the predicted series vanishes. This obstruction implies that every fixed-order Taylor truncation of the homogeneous Rogers--Szegő polynomial at $q=1$ fails to complete five powers of linear forms to the predicted series in all sufficiently large degrees. As a fixed-degree counterpart, we use exact certificates to prove the conjecture for any number of general quintics or septics; together with known results, this establishes the equal-degree conjecture in four variables for $1\le d\le 9$.

arXiv comment11 pages. v4 adds an obstruction for six forms with high-multiplicity linear factors and an application to Rogers-Szego truncations; includes exact certificates for quintics and septics