The initial ideal of generic sequences and Fröberg's Conjecture

Abstract

Let $K$ be an infinite field and let $I=(f_1,\cdots ,f_r)$ be an ideal in the polynomial ring $R=K[x_1,\cdots ,x_n]$ generated by generic forms of degrees $d_1,\cdots ,d_r$. A longstanding conjecture by Fröberg predicts the shape of the Hilbert function of $R/I.$ In 2010 Pardue stated a conjecture on the initial ideal of $n$ generic forms with respect to the deg-revlex order and he proved that it is equivalent to Fröberg's Conjecture. We study Pardue's Conjecture and we prove it under suitable conditions on the degrees of the forms. This yields a partial solution to Fröberg's Conjecture in the case $r\le n+2$ over an infinite field of any characteristic.

15 pages