稀疏铺砌拟阵的$Z$-多项式实根性及均匀拟阵的严格交错性
Real-rootedness of the $Z$-polynomials of sparse paving matroids and strict interlacing for uniform matroids
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中文总结 AI 辅助
本文证明稀疏铺砌拟阵的$Z$-多项式仅有负实零点，证实实根性猜想，并证明固定余秩下均匀拟阵的$Z$-多项式严格交错，方法是通过$\gamma$-多项式转移性质。
中文摘要 AI 辅助
我们证明了每个稀疏铺砌拟阵的$Z$-多项式仅有负实零点，从而证实了Proudfoot、Xu和Young关于此类拟阵的实根性猜想。我们还证明了，对于每个固定的正余秩，连续秩的均匀拟阵的$Z$-多项式严格交错。我们的方法是证明稀疏铺砌拟阵对应的$\gamma$-多项式的实根性以及均匀拟阵的严格交错性，然后将这两个性质转移到相应的$Z$-多项式上。
英文摘要
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 $γ$-polynomials for sparse paving matroids and their strict interlacing for uniform matroids, and then transfer both properties to the corresponding $Z$-polynomials.
