Recently, Ficarra and Sgroi initiated the study of v-numbers of powers of graded ideals. They proved that for a graded ideal I in a polynomial ring S, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textrm{v}(I^k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>v</mtext> <mo stretchy="false">(</mo> <msup> <mi>I</mi> <mi>k</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a linear function in k for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$k>>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>></mo> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Later, Ficarra conjectured that if I is a monomial ideal with linear powers, then <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textrm{v}(I^k)=\alpha (I)k-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>v</mtext> <mrow> <mo stretchy="false">(</mo> <msup> <mi>I</mi> <mi>k</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>α</mi> <mrow> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$k\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha (I)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> denotes the initial degree of I. In this paper, we generalize this conjecture for graded ideals. We prove this conjecture for several classes of graded ideals: principal ideals, ideals I with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textrm{depth}(S/I)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>depth</mtext> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">/</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, cover ideals of graphs, t-path ideals, monomial ideals generated in degree 2, edge ideals of weighted oriented graphs. We reduce the conjecture for several classes of graded ideals (including square-free monomial ideals) by showing that it is enough to prove the conjecture for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$k=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> only. Furthermore, we explicitly derive the asymptotic linear function <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\textrm{v}(I^k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>v</mtext> <mo stretchy="false">(</mo> <msup> <mi>I</mi> <mi>k</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for the edge ideals of connected graphs. Also, we define the stability index of the <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\textrm{v}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>v</mtext> </math></EquationSource> </InlineEquation>-numbers for graded ideals and investigate the stability index for edge ideals of graphs.

Asymptotic Behaviour and Stability Index of v-Numbers of Graded Ideals

期刊论文
发表日期：

2026年5月2日

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Prativa Biswas,
Mousumi Mandal,
Kamalesh Saha

摘要

Recently, Ficarra and Sgroi initiated the study of v-numbers of powers of graded ideals. They proved that for a graded ideal I in a polynomial ring S, $\textrm{v}(I^k)$ $v (I^{k})$ is a linear function in k for $k>>0$ $k > > 0$ . Later, Ficarra conjectured that if I is a monomial ideal with linear powers, then $\textrm{v}(I^k)=\alpha (I)k-1$ $v (I^{k}) = α (I) k - 1$ for all $k\ge 1$ $k \geq 1$ , where $\alpha (I)$ $α (I)$ denotes the initial degree of I. In this paper, we generalize this conjecture for graded ideals. We prove this conjecture for several classes of graded ideals: principal ideals, ideals I with $\textrm{depth}(S/I)=0$ $depth (S / I) = 0$ , cover ideals of graphs, t-path ideals, monomial ideals generated in degree 2, edge ideals of weighted oriented graphs. We reduce the conjecture for several classes of graded ideals (including square-free monomial ideals) by showing that it is enough to prove the conjecture for $k=1$ $k = 1$ only. Furthermore, we explicitly derive the asymptotic linear function $\textrm{v}(I^k)$ $v (I^{k})$ for the edge ideals of connected graphs. Also, we define the stability index of the $\textrm{v}$ $v$ -numbers for graded ideals and investigate the stability index for edge ideals of graphs.