The initial ideal of generic sequences and Fr\"{o}berg's Conjecture

TL;DR

This paper investigates Pardue's Conjecture regarding the initial ideal of generic sequences and its equivalence to Fr"{o}berg's Conjecture, providing partial proofs under certain degree conditions and advancing understanding of Hilbert functions of generic ideals.

Contribution

The paper proves Pardue's Conjecture under specific degree conditions, offering a partial resolution to Fr"{o}berg's Conjecture for cases where r ≤ n+2.

Findings

Pardue's Conjecture holds under certain degree conditions.

Partial proof of Fr"{o}berg's Conjecture for r ≤ n+2.

Results apply over any characteristic field.

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 Fr\"{o}berg 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 Fr\"{o}berg'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 Fr\"{o}berg's Conjecture in the case $r \leq n+2$ over an infinite field of any characteristic.