单峰性与某些二元形式洛朗级数
Unimodality and certain bivariate formal Laurent series
AI总结：
本文研究非负系数二元形式洛朗级数的单峰性，证明其构成交换半环，并据此解决Andrews的开放问题，建立高斯多项式、分拆秩统计量及Hilbert方案相关序列的单峰性。
摘要：
本文研究了具有非负系数的某些二元形式洛朗级数的单峰性和严格单峰性。我们证明了这些二元形式洛朗级数的集合在形式洛朗级数的加法和乘法运算下构成交换半环。利用这一结果，我们建立了涉及高斯多项式和某些细化颜色分拆的序列的单峰性。特别地，我们解决了Andrews提出的关于广义高斯多项式单峰性的一个开放问题，并建立了平面分拆的一个统计量的单峰性结果。我们还建立了分拆理论中秩统计量的许多单峰性结果，包括Andrews研究的凹和凸组合的秩统计量，以及Kim-Lim-Lovejoy研究的某些单峰序列。此外，我们建立了某些点的Hilbert方案的Betti数和Gromov-Witten不变量的单峰性。
英文摘要：
In this paper, we examine the unimodality and strict unimodality of certain formal bivariate Laurent series with non-negative coefficients. We show that the sets of these formal bivariate Laurent series form commutative semirings under the operations of addition and multiplication of formal Laurent series. This result is used to establish the unimodality of sequences involving Gauss polynomials and certain refined color partitions. In particular, we solve an open problem posed by Andrews on the unimodality of generalized Gauss polynomials and establish an unimodal result for a statistic of plane partitions. We also establish many unimodal results for rank statistics in partition theory, including the rank statistics of concave and convex compositions studied by Andrews, as well as certain unimodal sequences studied by Kim-Lim-Lovejoy. Additionally, we establish the unimodality of the Betti numbers and Gromov-Witten invariants of certain Hilbert schemes of points.
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