An upper bound is given on the smallest number <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1450_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$n_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>n</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> such that the maximal generating degree <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1450_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$d(I^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <msup> <mi>I</mi> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of a monomial ideal I in a polynomial ring becomes a linear function of n for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1450_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge n_0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <msub> <mi>n</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. The bound depends on d(I) and is an exponential function of the number of variables. In the case of two variables, we show that <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1450_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">$4d(I) - 8$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>4</mn> <mi>d</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mn>8</mn> </mrow> </math></EquationSource> </InlineEquation> is an upper bound.

Stability of maximal generating degrees of powers of monomial ideals

期刊论文
发表日期：

2025年9月3日

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Le Tuan Hoa

摘要

An upper bound is given on the smallest number $n_0$ $n_{0}$ such that the maximal generating degree $d(I^n)$ $d (I^{n})$ of a monomial ideal I in a polynomial ring becomes a linear function of n for $n\ge n_0$ $n \geq n_{0}$ . The bound depends on d(I) and is an exponential function of the number of variables. In the case of two variables, we show that $4d(I) - 8$ $4 d (I) - 8$ is an upper bound.