Realizable (reg, deg h)-Pairs for Cover Ideals via Independence Polynomials
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| Format: | Preprint |
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2026
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| author | Biermann, Jennifer Chau, Trung Kara, Selvi O'Keefe, Augustine Skelton, Joseph Castillo, Gabriel Sosa Vien, Dalena |
| author_facet | Biermann, Jennifer Chau, Trung Kara, Selvi O'Keefe, Augustine Skelton, Joseph Castillo, Gabriel Sosa Vien, Dalena |
| contents | Let $G$ be a finite simple graph on $n$ vertices and set $R=\Bbbk[x_1,\dots,x_n]$, with edge ideal $I(G)$ and cover ideal $J(G)$. We give an explicit description of the $h$-polynomial of $R/J(G)$, in a form that extends to the Alexander dual of any squarefree monomial ideal. We then express $\textrm{deg } h_{R/I(G)}(t)$ and $\textrm{deg } h_{R/J(G)}(t)$ in terms of the independence polynomial $P_G(x)=\sum_{i\ge 0} g_i x^i$ via an invariant $M(G)$, the multiplicity of $x=-1$ as a root of $P_G(x)$. In particular, we prove \[\textrm{deg } h_{R/I(G)}(t)=α(G)-M(G) \qquad\text{and}\qquad \textrm{deg } h_{R/J(G)}(t)=n-2-M(G), \] where $α(G)$ is the independence number of $G$. As a corollary, $M(G)$ is the additive inverse of the $\mathfrak{a}$-invariants of $R/I(G)$ and $R/J(G)$. We develop recursions and closed formulas for $M(G)$ for broad graph families, and use them to analyze which (reg, deg h)-pairs occur for cover ideals within chordal classes, including explicit constructions realizing extremal behavior. We conclude with a conjectural bound on $\left|\textrm{reg }(R/J(G))-\textrm{deg } h_{R/J(G)}(t)\right|$ for connected graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_10376 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Realizable (reg, deg h)-Pairs for Cover Ideals via Independence Polynomials Biermann, Jennifer Chau, Trung Kara, Selvi O'Keefe, Augustine Skelton, Joseph Castillo, Gabriel Sosa Vien, Dalena Commutative Algebra Combinatorics 13D02, 13D40, 13C70, 05C31 Let $G$ be a finite simple graph on $n$ vertices and set $R=\Bbbk[x_1,\dots,x_n]$, with edge ideal $I(G)$ and cover ideal $J(G)$. We give an explicit description of the $h$-polynomial of $R/J(G)$, in a form that extends to the Alexander dual of any squarefree monomial ideal. We then express $\textrm{deg } h_{R/I(G)}(t)$ and $\textrm{deg } h_{R/J(G)}(t)$ in terms of the independence polynomial $P_G(x)=\sum_{i\ge 0} g_i x^i$ via an invariant $M(G)$, the multiplicity of $x=-1$ as a root of $P_G(x)$. In particular, we prove \[\textrm{deg } h_{R/I(G)}(t)=α(G)-M(G) \qquad\text{and}\qquad \textrm{deg } h_{R/J(G)}(t)=n-2-M(G), \] where $α(G)$ is the independence number of $G$. As a corollary, $M(G)$ is the additive inverse of the $\mathfrak{a}$-invariants of $R/I(G)$ and $R/J(G)$. We develop recursions and closed formulas for $M(G)$ for broad graph families, and use them to analyze which (reg, deg h)-pairs occur for cover ideals within chordal classes, including explicit constructions realizing extremal behavior. We conclude with a conjectural bound on $\left|\textrm{reg }(R/J(G))-\textrm{deg } h_{R/J(G)}(t)\right|$ for connected graphs. |
| title | Realizable (reg, deg h)-Pairs for Cover Ideals via Independence Polynomials |
| topic | Commutative Algebra Combinatorics 13D02, 13D40, 13C70, 05C31 |
| url | https://arxiv.org/abs/2602.10376 |