A palindromicity criterion for the $h$-polynomials of bipartite edge rings
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917540082483200 |
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| author | Hatasa, Yuta |
| author_facet | Hatasa, Yuta |
| contents | We study a symmetry problem for the $h$-polynomials of edge rings of bipartite graphs. Let $G$ be a bipartite graph and write $h(\mathbb{k}[G];t)=h_0+h_1t+\cdots+h_st^s$. We prove that if $\Bbbk[G]$ is pseudo-Gorenstein and $h_1=h_{s-1}$, then $\Bbbk[G]$ is Gorenstein. Equivalently, under these assumptions the $h$-polynomial of $\Bbbk[G]$ is palindromic. The proof treats the $2$-connected case first by translating the numerical condition $h_1=h_{s-1}$ into a tight-separation condition for non-edges, and then passes to arbitrary bipartite graphs using the block decomposition. We also construct a blockwise Gorenstein closure, obtained by adjoining all non-edges not separated by tight acceptable sets, and show that this construction preserves the next-to-leading coefficient of the $h$-polynomial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26865 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A palindromicity criterion for the $h$-polynomials of bipartite edge rings Hatasa, Yuta Commutative Algebra Combinatorics Primary 13F65, Secondary 13D40, 05C25, 52B20 We study a symmetry problem for the $h$-polynomials of edge rings of bipartite graphs. Let $G$ be a bipartite graph and write $h(\mathbb{k}[G];t)=h_0+h_1t+\cdots+h_st^s$. We prove that if $\Bbbk[G]$ is pseudo-Gorenstein and $h_1=h_{s-1}$, then $\Bbbk[G]$ is Gorenstein. Equivalently, under these assumptions the $h$-polynomial of $\Bbbk[G]$ is palindromic. The proof treats the $2$-connected case first by translating the numerical condition $h_1=h_{s-1}$ into a tight-separation condition for non-edges, and then passes to arbitrary bipartite graphs using the block decomposition. We also construct a blockwise Gorenstein closure, obtained by adjoining all non-edges not separated by tight acceptable sets, and show that this construction preserves the next-to-leading coefficient of the $h$-polynomial. |
| title | A palindromicity criterion for the $h$-polynomials of bipartite edge rings |
| topic | Commutative Algebra Combinatorics Primary 13F65, Secondary 13D40, 05C25, 52B20 |
| url | https://arxiv.org/abs/2605.26865 |