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<p>Let <i>U</i><sub><i>n,d</i></sub> denote the uniform matroid of rank <i>d</i> on <i>n</i> elements. We obtain some recurrence relations satisfied by Speyer’s <i>g</i>-polynomials <i>g</i><Emphasis Type="ItalicSmallCaps">u</Emphasis><sub><i>n,d</i></sub>(<i>t</i>) of <i>U</i><sub><i>n,d</i></sub>. Based on these recurrence relations, we prove that the polynomial <i>g</i><Emphasis Type="ItalicSmallCaps">u</Emphasis><sub><i>n,d</i></sub>(<i>t</i>) has only real zeros for any <i>n</i> − 1 ≥ <i>d</i> ≥ 1. Furthermore, we show that the coefficient of <i>g</i><Emphasis Type="ItalicSmallCaps">u</Emphasis><sub><i>n</i>,[<i>n</i>/2]</sub>(<i>t</i>) is asymptotically normal by local and central limit theorems.</p>

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Analytic Properties of Speyer’s g-polynomial of Uniform Matroids

  • 期刊论文

  • 发表日期：

    2026年4月15日

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  • Rong Zhang,
  • James J. Y. Zhao

摘要

Let Un,d denote the uniform matroid of rank d on n elements. We obtain some recurrence relations satisfied by Speyer’s g-polynomials gun,d(t) of Un,d. Based on these recurrence relations, we prove that the polynomial gun,d(t) has only real zeros for any n − 1 ≥ d ≥ 1. Furthermore, we show that the coefficient of gun,[n/2](t) is asymptotically normal by local and central limit theorems.