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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.
Let $A = K[X_1,ldots, X_d]$ and let $I$, $J$ be monomial ideals in $A$. Let $I_n(J) = (I^n colon J^infty)$ be the $n^{th}$ symbolic power of $I$ wrt $J$. It is easy to see that the function $f^I_J(n) = e_0(I_n(J)/I^n)$ is of quasi-polynomial type, s
Given a number $q$, we construct a monomial ideal $I$ with the property that the function which describes the number of generators of $I^k$ has at least $q$ local maxima.
Let $S=K[x_1,ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $Isubset S$ a squarefree monomial ideal. In the present paper we are interested in the monomials $u in S$ belonging to the socle $Soc(S/I^{k})$ of $S/I^{k}$, i.e.,
We investigate the rational powers of ideals. We find that in the case of monomial ideals, the canonical indexing leads to a characterization of the rational powers yielding that symbolic powers of squarefree monomial ideals are indeed rational power
This paper is concerned with the question of whether geometric structures such as cell complexes can be used to simultaneously describe the minimal free resolutions of all powers of a monomial ideal. We provide a full answer in the case of square-fre