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PreprintCC BY 4.0arXiv

A palindromicity criterion for the h-polynomials of bipartite edge rings

  • Yuta Hatasa
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Abstract

We study a symmetry problem for the h-polynomials of edge rings of bipartite graphs. Let G be a bipartite graph and write h(k[G];t)=h0+h1t+⋯+hsts. We prove that if Bbbk[G] is pseudo-Gorenstein and h1=hs−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 h1=hs−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 minimal 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.

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Cite

BibTeXRIS
BibTeX
@misc{hatasa2026palindromicity,
  title = {{A palindromicity criterion for the $h$-polynomials of bipartite edge rings}},
  author = {Yuta Hatasa},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.26865},
  eprint = {2605.26865},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.26865},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A palindromicity criterion for the h-polynomials of bipartite edge rings
AU  - Yuta Hatasa
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2605.26865
UR  - https://arxiv.org/abs/2605.26865
AB  - We study a symmetry problem for the h-polynomials of edge rings of bipartite graphs. Let G be a bipartite graph and write h(k[G];t)=h_0+h_1t+⋯+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 minimal 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.
ER  -