Simon Conjecture and the $\text{v}$-number of monomial ideals

TL;DR

This paper extends the Simon conjecture to all monomial ideals using polarization, and explores the behavior of the v-number of powers of monomial ideals, providing evidence for a specific formula in various classes.

Contribution

It generalizes the Simon conjecture to monomial ideals and investigates the v-number behavior of their powers, especially those with linear powers.

Findings

If Simon conjecture holds and all powers have linear quotients, then b is in {-1,0}.

For equigenerated monomial ideals with linear powers, v(I^k) = α(I)k - 1 for all k ≥ 1.

The conjecture is verified for specific classes like edge ideals with linear resolution, polymatroidal ideals, and Hibi ideals.

Abstract

Let $I\subset S$ be a graded ideal of a standard graded polynomial ring $S$ with coefficients in a field $K$, and let $\text{v}(I)$ be the $\text{v}$-number of $I$. In previous work, we showed that for any graded ideal $I\subset S$ generated in a single degree, then $\text{v}(I^k)=\alpha(I)k+b$, for all $k\gg0$, where $\alpha(I)$ is the initial degree of $I$ and $b$ is a suitable integer. In the present paper, using polarization, we extend Simon conjecture to any monomial ideal. As a consequence, if Simon conjecture holds, and all powers of $I$ have linear quotients, then $b\in\{-1,0\}$. This fact suggest that if $I$ is an equigenerated monomial ideal with linear powers, then $\text{v}(I^k)=\alpha(I)k-1$, for all $k\ge1$. We verify this conjecture for monomial ideals with linear powers having $\text{depth}S/I=0$, edge ideals with linear resolution, polymatroidal ideals, and Hibi ideals.