Hilbert functions of filtered graphical zonotopal algebras
Abstract
The external zonotopal algebra of a graph is generated by nilpotent vertex elements Xv and determines the graphical matroid after deleting loops. Replacing these generators by f(Xv), for a power series f(u), and filtering by word length gives a family of filtered algebras. We extend filtered graph reconstruction from simple graphs to loopless multigraphs without isolated vertices when the linear and quadratic coefficients of f are nonzero. Nevertheless, even the family of Hilbert functions indexed by all polynomials does not determine the graph: two nonisomorphic ten-vertex trees have the same Hilbert function for every f, answering a question of Nenashev and the first author. For forests and series whose first nonconstant term has degree k≥ 2, we give a monomial presentation indexed by partial score vectors whose entries are divisible by k. For normalized nondegenerate functions, a boundary-state calculus gives a common finite parameter partition for all trees of a fixed maximum degree. In maximum degrees three and four, exact symbolic calculations identify these partitions by the vanishing of specified coefficients of f−1. For functions with nonzero quadratic coefficient, the same calculus gives explicit recurrences for attached or connecting paths. Symmetry reductions, including commuting actions of sl2 and the symmetric group on the leaves of a star, determine the generic Hilbert series of stars and two-vertex multigraphs and also apply to parallel edges and leaves in larger graphs. For cycles, we determine the highest-degree term and the first three nonconstant coefficients in the stable range. A weighted independent-set formula describes the trace sequence in the remaining conjecture.
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BibTeX
@misc{shapiro2026hilbert,
title = {{Hilbert functions of filtered graphical zonotopal algebras}},
author = {Boris Shapiro and Ilya Smirnov and Arkady Vaintrob},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38469},
eprint = {2610.38469},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38469},
}
RIS
TY - UNPB
TI - Hilbert functions of filtered graphical zonotopal algebras
AU - Boris Shapiro
AU - Ilya Smirnov
AU - Arkady Vaintrob
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38469
UR - https://arxiv.org/abs/2610.38469
AB - The external zonotopal algebra of a graph is generated by nilpotent vertex elements X_v and determines the graphical matroid after deleting loops. Replacing these generators by f(X_v), for a power series f(u), and filtering by word length gives a family of filtered algebras. We extend filtered graph reconstruction from simple graphs to loopless multigraphs without isolated vertices when the linear and quadratic coefficients of f are nonzero. Nevertheless, even the family of Hilbert functions indexed by all polynomials does not determine the graph: two nonisomorphic ten-vertex trees have the same Hilbert function for every f, answering a question of Nenashev and the first author. For forests and series whose first nonconstant term has degree k≥ 2, we give a monomial presentation indexed by partial score vectors whose entries are divisible by k. For normalized nondegenerate functions, a boundary-state calculus gives a common finite parameter partition for all trees of a fixed maximum degree. In maximum degrees three and four, exact symbolic calculations identify these partitions by the vanishing of specified coefficients of f^−1. For functions with nonzero quadratic coefficient, the same calculus gives explicit recurrences for attached or connecting paths. Symmetry reductions, including commuting actions of sl_2 and the symmetric group on the leaves of a star, determine the generic Hilbert series of stars and two-vertex multigraphs and also apply to parallel edges and leaves in larger graphs. For cycles, we determine the highest-degree term and the first three nonconstant coefficients in the stable range. A weighted independent-set formula describes the trace sequence in the remaining conjecture.
ER -