Powers of generic ideals and the weak Lefschetz property for powers of some monomial complete intersections

TL;DR

This paper investigates the Hilbert functions of powers of ideals generated by forms of degree d in polynomial rings, focusing on cases with more generators than variables and exploring the Weak Lefschetz property for specific monomial ideals.

Contribution

It extends understanding of Hilbert functions and the Weak Lefschetz property for powers of ideals with more generators than variables, especially for the case r=n+1.

Findings

Characterized the minimal Hilbert function for powers of certain ideals.

Analyzed the Weak Lefschetz property for ideals generated by variable powers.

Provided new bounds and conditions for these algebraic properties.

Abstract

Given an ideal $I=(f_1,\ldots,f_r)$ in $\mathbb C[x_1,\ldots,x_n]$ generated by forms of degree $d$, and an integer $k>1$, how large can the ideal $I^k$ be, i.e., how small can the Hilbert function of $\mathbb C[x_1,\ldots,x_n]/I^k$ be? If $r\le n$ the smallest Hilbert function is achieved by any complete intersection, but for $r>n$, the question is in general very hard to answer. We study the problem for $r=n+1$, where the result is known for $k=1$. We also study a closely related problem, the Weak Lefschetz property, for $S/I^k$, where $I$ is the ideal generated by the $d$'th powers of the variables.