In the literature, two distinct notions of symbolic powers of an ideal exist: one is based on the set of all minimal primes of I, while the other involves the entire set of associated primes of the ideal I. Using the first definition, Grisalde et al. (Normally torsion-free edge ideals of weighted oriented graphs, Comm. Algebra 52 (2024), no. 4, 1672–1685) classified all edge ideals of weighted oriented graphs for which the symbolic and ordinary powers coincide. In this article, we explore the same question under the second definition of symbolic powers. Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1842_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">D</mi> </math></EquationSource> </InlineEquation> be a weighted oriented graph, and let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1842_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$I(\mathcal {D})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">D</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> denote its edge ideal. We demonstrate that all symbolic and ordinary powers of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1842_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$I(\mathcal {D})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">D</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> coincide when <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1842_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">D</mi> </math></EquationSource> </InlineEquation> is a weighted oriented tree of a specific class. Finally, we provide necessary and sufficient conditions for the equality of ordinary and symbolic powers of naturally oriented paths.

Ordinary and Symbolic Powers of Edge Ideals of Weighted Oriented Graphs

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发表日期：

2025年4月8日

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Arindam Banerjee,
Kanoy Kumar Das,
S. Selvaraja

摘要

In the literature, two distinct notions of symbolic powers of an ideal exist: one is based on the set of all minimal primes of I, while the other involves the entire set of associated primes of the ideal I. Using the first definition, Grisalde et al. (Normally torsion-free edge ideals of weighted oriented graphs, Comm. Algebra 52 (2024), no. 4, 1672–1685) classified all edge ideals of weighted oriented graphs for which the symbolic and ordinary powers coincide. In this article, we explore the same question under the second definition of symbolic powers. Let $\mathcal {D}$ $D$ be a weighted oriented graph, and let $I(\mathcal {D})$ $I (D)$ denote its edge ideal. We demonstrate that all symbolic and ordinary powers of $I(\mathcal {D})$ $I (D)$ coincide when $\mathcal {D}$ $D$ is a weighted oriented tree of a specific class. Finally, we provide necessary and sufficient conditions for the equality of ordinary and symbolic powers of naturally oriented paths.