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

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