Magic positivity of Snapper polynomials for matroids
Abstract
In 1959, Snapper showed that the Euler characteristic of the tensor powers of a line bundle on a normal projective scheme is a polynomial, later named the \emph{Snapper polynomial}. Positivity of coefficients of Snapper polynomials implies various notions of positivity of line bundles, which we study through the lens of magic positivity and real-rootedness. We introduce zonotopal classes in the Grothendieck K-ring of vector bundles of the toric variety for any loopless matroid, and prove that their Snapper polynomials are magic positive. Our proof realizes such a Snapper polynomial as a weighted independence polynomial of the Dilworth truncation along certain lines of the matroid. As a consequence, their coefficients are positive, and their h∗-polynomials are real-rooted. In the realizable case, this polynomial is the multigraded Hilbert polynomial of the wonderful variety embedded in a product of projective lines. We introduce analogous line bundles on the Deligne--Mumford--Knudsen moduli space ℳ̅0,n and prove that their Snapper polynomials are magic positive. For cotangent line bundles whose first Chern classes are distinct ψ-classes, which are not zonotopal, we nonetheless prove that their h∗-polynomials are real-rooted, whereas their Snapper polynomials are magic positive if and only if nleqslant7. More generally, we introduce saturated and weakly saturated K-classes of matroids, which furnish a sufficient and a necessary condition for magic positivity of Snapper polynomials in terms of their dragon Hall--Rado polymatroids.
Related
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BibTeX
@misc{li2026magic,
title = {{Magic positivity of Snapper polynomials for matroids}},
author = {Shiyue Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24917},
eprint = {2609.24917},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24917},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Magic positivity of Snapper polynomials for matroids
AU - Shiyue Li
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24917
UR - https://arxiv.org/abs/2609.24917
AB - In 1959, Snapper showed that the Euler characteristic of the tensor powers of a line bundle on a normal projective scheme is a polynomial, later named the \emph{Snapper polynomial}. Positivity of coefficients of Snapper polynomials implies various notions of positivity of line bundles, which we study through the lens of magic positivity and real-rootedness. We introduce zonotopal classes in the Grothendieck K-ring of vector bundles of the toric variety for any loopless matroid, and prove that their Snapper polynomials are magic positive. Our proof realizes such a Snapper polynomial as a weighted independence polynomial of the Dilworth truncation along certain lines of the matroid. As a consequence, their coefficients are positive, and their h^∗-polynomials are real-rooted. In the realizable case, this polynomial is the multigraded Hilbert polynomial of the wonderful variety embedded in a product of projective lines. We introduce analogous line bundles on the Deligne--Mumford--Knudsen moduli space ℳ̅_0,n and prove that their Snapper polynomials are magic positive. For cotangent line bundles whose first Chern classes are distinct ψ-classes, which are not zonotopal, we nonetheless prove that their h^∗-polynomials are real-rooted, whereas their Snapper polynomials are magic positive if and only if nleqslant7. More generally, we introduce saturated and weakly saturated K-classes of matroids, which furnish a sufficient and a necessary condition for magic positivity of Snapper polynomials in terms of their dragon Hall--Rado polymatroids.
ER -