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arXiv · 2606.22933

Prescribed Initial Behavior of $\mu(I^k)$

Reza Abdolmaleki·Shinya Kumashiro
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Abstract

It is well known that for every graded ideal $I$, the numbers of minimal generators $\mu(I^k)$ of its powers of $I$ are eventually increasing. However, its initial behavior can be surprisingly flexible. We prove that any prescribed finite pattern of increases, decreases, and equalities can occur among the first differences $\mu(I^{k+1})-\mu(I^k)$ of a suitable monomial ideal $I$ in $K[x,y]$. This provides a broad positive answer to a previously posed sign-realization problem and unifies several known constructions exhibiting unusual behavior of $\mu(I^k)$. Moreover, as a consequence, analogous sign-realization results are obtained for the index of reducibility of powers of $\mathfrak m$-primary ideals in two-dimensional regular local rings.

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Reza Abdolmaleki, Shinya Kumashiro. 2026-06-22. Prescribed Initial Behavior of $\mu(I^k)$. https://arxiv.org/abs/2606.22933

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