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.

翻译：2013年，Pak与Panova基于Young表组合学及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）的严格单峰性质，并指出该性质对所有$\ell,m\geq 8$及若干其他情形成立。我们提出一种基于计算机代数的不同方法：建立这些多项式系数的封闭形式，进而利用柱形代数分解精确判定严格单峰性成立的系数范围。该策略使我们能够处理问题的推广形式，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与Zanello猜想中两个附加情形的证明。