bibkey: zhangzhao2024speyer
authors: Rong Zhang, James Jing Yu Zhao
year: 2024
title: Analytic properties of Speyer’s g-polynomial of uniform matroids
doi: 10.1007/s10114-026-4502-5
url: https://arxiv.org/abs/2408.15506v1
claim: Proposition 1.1 gives the uniform-matroid polynomial; Theorem 4.2 gives its normality criterion; Conjecture 4.8 asks for central and local normal limits when rank diverges.
strata_touched:
  - D5/S3/Analytic/Probability/BernoulliTriangularLimits
  - D5/S3/Combinatorics/Matroids/SpeyerUniformNormality
license: citation-only
triage: anchor

Analytic properties of Speyer’s g-polynomial of uniform matroids

Verified locator

DOI: 10.1007/s10114-026-4502-5

Source: https://arxiv.org/abs/2408.15506v1

Source statements

Proposition 1.1, page 1:

For any n ≥ 2 and 1 ≤ d ≤ n − 1, the g-polynomial of the uniform matroid Uₙ,ᵈ is given by $g_{U_{n,d}}(t)={\sum }_{i=1}^{\min (d,n-d)}\frac{(n-i-1)!}{(d-i)!(n-d-i)!(i-1)!}{t}^i$.

Theorem 4.2, page 9, attributes the criterion to Bender’s Theorem 2:

If ${\sigma }_n^2\to +\infty$ as $n\to +\infty$, then the coefficient of $f_n(t)$ is asymptotically normal with mean ${\mu }_n$ and variance ${\sigma }_n^2$ by local and central limit theorems.

Conjecture 4.8, page 13:

If d = d(n) is a real function of n such that d(n) → +∞ as n → +∞, then the coefficient of Speyer’s g-polynomial gₙ,ᵈ₍ₙ₎(t), namely Sᵢ(n,d(n)), is asymptotically normal by local and central limit theorems.

Encoding and scope

Rank is integer-valued, and the standing half-rank restriction is required eventually. The polynomial is the literal factorial sum of Proposition 1.1. The derivative ratios are the mean and variance of Theorem 4.2. The epsilon-form limits quantify uniformly over every real threshold. Negative integer coefficient indices have probability zero. The existing Mathlib owners gaussianPDFReal 0 1 and cdf (gaussianReal 0 1) are respectively the source’s standard Gaussian density and its integral over Set.Iic.

The source proves the balanced half-rank case in Theorem 4.1 and ends with Conjecture 4.8. It makes no bounded-rank claim in that conjecture.

Literature boundary

The journal version is Acta Mathematica Sinica, English Series 42, 1087–1098. Its full text remains ASSUMED-UNVERIFIED; the arXiv v1 statement is the verified target.