Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1436_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="345" /> </InlineMediaObject> <EquationSource Format="TEX">$J_{n,m} = (x_1\cdots x_{m},x_2 \cdots x_{m+1},\ldots ,x_{n}x_1\cdots x_{m-1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>J</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>⋯</mo> <msub> <mi>x</mi> <mi>m</mi> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>⋯</mo> <msub> <mi>x</mi> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>⋯</mo> <msub> <mi>x</mi> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the m-path ideal of a cycle of length <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1436_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n \ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> over the polynomial ring <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1436_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="123" /> </InlineMediaObject> <EquationSource Format="TEX">$S = \textrm{k}[x_1,\ldots ,x_n]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>=</mo> <mtext>k</mtext> <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>. We provide formulae for all the Betti numbers of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1436_Article_IEq4.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">$J_{n,m}^t$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>J</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mrow> <mi>t</mi> </msubsup> </math></EquationSource> </InlineEquation> for all positive integers t when <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1436_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$m = n-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1436_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$m = n-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.

Betti numbers of powers of path ideals of cycles

期刊论文
发表日期：

2025年6月16日

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Silviu Bălănescu,
Mircea Cimpoeaş,
Thanh Vu

摘要

Let $J_{n,m} = (x_1\cdots x_{m},x_2 \cdots x_{m+1},\ldots ,x_{n}x_1\cdots x_{m-1})$ $J_{n, m} = (x_{1} \dots x_{m}, x_{2} \dots x_{m + 1}, \dots, x_{n} x_{1} \dots x_{m - 1})$ be the m-path ideal of a cycle of length $n \ge 5$ $n \geq 5$ over the polynomial ring $S = \textrm{k}[x_1,\ldots ,x_n]$ $S = k [x_{1}, \dots, x_{n}]$ . We provide formulae for all the Betti numbers of $J_{n,m}^t$ $J_{n, m}^{t}$ for all positive integers t when $m = n-1$ $m = n - 1$ or $m = n-2$ $m = n - 2$ .