Real-rootedness of the $Z$-polynomials of sparse paving matroids and strict interlacing for uniform matroids

Matthew H. Y. Xie; Philip B. Zhang

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arXiv · 2609.07636

Real-rootedness of the $Z$-polynomials of sparse paving matroids and strict interlacing for uniform matroids

Matthew H. Y. Xie·Philip B. Zhang

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Abstract

We prove that the $Z$-polynomial of every sparse paving matroid has only negative real zeros, confirming the real-rootedness conjecture of Proudfoot, Xu, and Young for this class. We also prove that, for each fixed positive corank, the $Z$-polynomials of uniform matroids in consecutive ranks strictly interlace. Our approach is to prove real-rootedness of the corresponding $\gamma$-polynomials for sparse paving matroids and their strict interlacing for uniform matroids, and then transfer both properties to the corresponding $Z$-polynomials.

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Matthew H. Y. Xie, Philip B. Zhang. 2026-09-07. Real-rootedness of the $Z$-polynomials of sparse paving matroids and strict interlacing for uniform matroids. https://arxiv.org/abs/2609.07636

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