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Asymptotic behaviour of the $\text{v}$-number of homogeneous ideals

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arxiv 2306.14243 v4 pith:726F4GQZ submitted 2023-06-25 math.AC

Antonino Ficarra , Emanuele Sgroi This is my paper · ORCID
classification math.AC
keywords textfunctiongradedalphaasymptoticbehaviouridealideals
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abstract

Let $I$ be a graded ideal of a standard graded polynomial ring $S$ with coefficients in a field $K$. The asymptotic behaviour of the $\text{v}$-number of the powers of $I$ is investigated. Natural lower and upper bounds which are linear functions in $k$ are determined for $\text{v}(I^k)$. We call $\text{v}(I^k)$ the $\text{v}$-function of $I$. We prove that $\text{v}(I^k)$ is a linear function in $k$ for $k$ large enough, of the form $\text{v}(I^k)=\alpha(I)k+b$, where $\alpha(I)$ is the initial degree of $I$, and $b\in\mathbb{Z}$ is a suitable integer. For this aim, we construct new blowup algebras associated to graded ideals. Finally, for a monomial ideal in two variables, we compute explicitly its $\text{v}$-function.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Comparing the $\mathrm{v}$-number and $h$-polynomials of edge ideals

    math.AC 2025-07 conditional novelty 7.0 of 10

    For edge ideals of connected graphs, the v-number can be arbitrarily larger or smaller than the degree of the h-polynomial, their sum is at most the number of vertices, and equality holds exactly for disjoint unions o...

  2. $\operatorname{v}$-numbers of integral closure filtrations of monomial ideals

    math.AC 2025-06 reject novelty 6.0 of 10

    For certain monomial ideals, the paper proposes explicit values and bounds for v-numbers of integral closure filtrations and shows they can be smaller than v-numbers of ordinary powers.

  3. Comparison of stability indices of powers of graded ideals

    math.AC 2025-05 accept novelty 6.0 of 10

    The paper proves that astab(I)=1 for every graded ideal in dimension two with vstab(I) arbitrary, and constructs ideals realizing every pair (astab(I), vstab(I))=(a,b) in higher dimension.

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