The saturation number of $\cb$-bounded stable monomial ideals and their powers

TL;DR

This paper computes the socle and saturation number of $eta$-bounded strongly stable ideals, providing explicit formulas and showing the quasi-linear behavior of saturation numbers for their powers, including Veronese type ideals.

Contribution

It introduces explicit formulas for the saturation number of $eta$-bounded strongly stable ideals and their powers, including Veronese type ideals, revealing their quasi-linear nature.

Findings

Explicit formulas for saturation numbers of $eta$-bounded strongly stable ideals.

Demonstration that saturation numbers of powers are quasi-linear.

Determination of the quasi-linear function for saturation numbers.

Abstract

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$. In this paper, we compute the socle of $\cb$-bounded strongly stable ideals and determine that the saturation number of strongly stable ideals and of equigenerated $\cb$-bounded strongly stable ideals. We also provide explicit formulas for the saturation number $\sat(I)$ of Veronese type ideals $I$. Using this formula, we show that $\sat(I^k)$ is quasi-linear from the beginning and we determine the quasi-linear function explicitly.