高斯多项式单峰性的统一方法
A Unified Approach to Unimodality of Gaussian Polynomials
AI总结：
提出基于计算机代数的新方法，通过闭式系数和柱形代数分解精确判定$q$-二项式系数的严格单峰性，并证明Stanley和Zanello猜想的两个额外情形。
摘要：
2013年，Pak和Panova基于Young tableau的组合学以及Kronecker系数的半群性质，证明了$q$-二项式系数$\binom{\ell+m}{m}_q$（作为$q$的多项式）的严格单峰性。他们证明该性质对所有$\ell,m\geq 8$以及少数其他情形成立。我们提出一种基于计算机代数的不同方法，其中我们建立这些多项式系数的闭式形式，然后使用柱形代数分解来精确识别严格单峰性成立时的系数范围。该策略使我们能够处理该问题的推广，例如证明具有更大间隔的单峰性或相关序列的单峰性。特别地，我们给出了Stanley和Zanello猜想中两个额外情形的证明。
英文摘要：
In 2013, Pak and Panova proved the strict unimodality property of $q$-binomial coefficients $\binom{\ell+m}{m}_q$ (as polynomials in $q$) based on the combinatorics of Young tableaux and the semigroup property of Kronecker coefficients. They showed it to be true for all $\ell,m\geq 8$ and a few other cases. We propose a different approach to this problem based on computer algebra, where we establish a closed form for the coefficients of these polynomials and then use cylindrical algebraic decomposition to identify exactly the range of coefficients where strict unimodality holds. This strategy allows us to tackle generalizations of the problem, e.g., to show unimodality with larger gaps or unimodality of related sequences. In particular, we present proofs of two additional cases of a conjecture by Stanley and Zanello.
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