Let K be a field, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1429_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="163" /> </InlineMediaObject> <EquationSource Format="TEX">$I\subset R=K[x_1,\dots ,x_n]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mo>⊂</mo> <mi>R</mi> <mo>=</mo> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1429_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="165" /> </InlineMediaObject> <EquationSource Format="TEX">$J\subset T=K[y_1,\dots ,y_m]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>J</mi> <mo>⊂</mo> <mi>T</mi> <mo>=</mo> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>y</mi> <mi>m</mi> </msub> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> be graded ideals. Set <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1429_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">$S=R\otimes _KT$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>=</mo> <mi>R</mi> <msub> <mo>⊗</mo> <mi>K</mi> </msub> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation> and let <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1429_Article_IEq6.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="100" /> </InlineMediaObject> <EquationSource Format="TEX">$L=IS+JS$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>=</mo> <mi>I</mi> <mi>S</mi> <mo>+</mo> <mi>J</mi> <mi>S</mi> </mrow> </math></EquationSource> </InlineEquation>. The behavior of the <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1429_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$v $</EquationSource> <EquationSource Format="MATHML"><math> <mi>v</mi> </math></EquationSource> </InlineEquation>-function <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1429_Article_IEq8.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$v (L^k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mi>k</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in terms of the <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1429_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$v $</EquationSource> <EquationSource Format="MATHML"><math> <mi>v</mi> </math></EquationSource> </InlineEquation>-functions <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1429_Article_IEq10.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$v (I^k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <msup> <mi>I</mi> <mi>k</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1429_Article_IEq11.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$v (J^k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <msup> <mi>J</mi> <mi>k</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is investigated. When I and J are monomial ideals, we describe <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1429_Article_IEq8.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$v (L^k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mi>k</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, giving an explicit formula involving the local <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1429_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$v $</EquationSource> <EquationSource Format="MATHML"><math> <mi>v</mi> </math></EquationSource> </InlineEquation>-numbers <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1429_Article_IEq14.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$v _{\mathfrak {p}}(I^k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>v</mi> <mi mathvariant="fraktur">p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>I</mi> <mi>k</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1429_Article_IEq15.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$v _{\mathfrak {q}}(J^k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>v</mi> <mi mathvariant="fraktur">q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>J</mi> <mi>k</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

The $v $-function of powers of sums of ideals

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发表日期：

2025年7月15日

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Antonino Ficarra,
Pedro Macias Marques

摘要

Let K be a field, $I\subset R=K[x_1,\dots ,x_n]$ $I \subset R = K [x_{1}, \dots, x_{n}]$ and $J\subset T=K[y_1,\dots ,y_m]$ $J \subset T = K [y_{1}, \dots, y_{m}]$ be graded ideals. Set $S=R\otimes _KT$ $S = R \otimes_{K} T$ and let $L=IS+JS$ $L = I S + J S$ . The behavior of the $v $ $v$ -function $v (L^k)$ $v (L^{k})$ in terms of the $v $ $v$ -functions $v (I^k)$ $v (I^{k})$ and $v (J^k)$ $v (J^{k})$ is investigated. When I and J are monomial ideals, we describe $v (L^k)$ $v (L^{k})$ , giving an explicit formula involving the local $v $ $v$ -numbers $v _{\mathfrak {p}}(I^k)$ $v_{p} (I^{k})$ and $v _{\mathfrak {q}}(J^k)$ $v_{q} (J^{k})$ .

The \(v \)-function of powers of sums of ideals

摘要