A study of $v$-number for some monomial ideals

TL;DR

This paper investigates the $v$-number of monomial ideals, providing formulas for specific graph-related ideals, explicit expressions for $rak{m}$-primary ideals, and bounds relating $v$-number to regularity, revealing new properties and bounds.

Contribution

It offers explicit formulas and bounds for the $v$-number of various monomial ideals, including those from graph theory, and explores its relationship with regularity.

Findings

Formulas for $v$-number of edge ideals of certain graphs.

An explicit expression for $v$-number of $rak{m}$-primary monomial ideals.

Upper bounds of $v$-number in terms of generator degrees.

Abstract

In this paper, we give formulas for $v$-number of edge ideals of some graphs like path, cycle, 1-clique sum of a path and a cycle, 1-clique sum of two cycles and join of two graphs. For an $\mathfrak{m}$-primary monomial ideal $I\subset S=K[x_1,\ldots,x_t]$, we provide an explicit expression of $v$-number of $I$, denoted by $v(I)$, and give an upper bound of $v(I)$ in terms of the degree of its generators. We show that for a monomial ideal $I$, $v(I^{n+1})$ is bounded above by a linear polynomial for large $n$ and for certain classes of monomial ideals, the upper bound is achieved for all $n\geq 1$. For $\mathfrak m$-primary monomial ideal $I$ we prove that $v(I)\leq$ reg$(S/I)$ and their difference can be arbitrarily large.