通过牛顿多面体研究簇代数中的递推公式、正性与多面体基
Recurrence formula, positivity and polytope basis in cluster algebras via Newton polytopes
AI总结：
本文通过牛顿多面体研究TSSS簇代数中F-多项式的性质，提出了簇变量Laurent表达式的递推构造，证明了多面体基的强正性，并构建了关键变量间的显式映射关系。
摘要：
本文研究了TSSS簇代数$\mathcal A$中$F$-多项式的牛顿多面体，并将其推广到由向量$h\in\Z^{n}$关联的多面体$N_{h}$组成的更大集合，以及由对应于$N_{h}$的多面体函数$ρ_{h}$组成的集合$\widehat{\mathcal{P}}$。主要贡献包括：(i) 通过簇变量的$g$-向量获得其在簇中Laurent表达式的递推构造；(ii) 证明当$\mathcal A$是具有主系数的簇代数时，$\widehat{\mathcal{P}}$中由Laurent多项式构成的子集$\mathcal{P}$是$\mathcal{U}(\A)$的一个强正$\Z Trop(Y)$-基，由某些普遍不可分解的Laurent多项式组成。对于任意半环$\mathbb P$上的簇代数$\mathcal A$，$\mathcal{P}$是$\mathcal{U(A)}$的中间簇子代数$\mathcal{I_P(A)}$的强正$\Z¶$-基。我们称$\mathcal P$为多面体基；(iii) 构造了相应的$F$-多项式、$g$-向量、$d$-向量和簇变量之间的显式映射，以刻画它们的关系。此外，我们分别给出了(i)、(ii)和(iii)的三个应用。
英文摘要：
In this paper, we study the Newton polytopes of $F$-polynomials in a TSSS cluster algebra $\mathcal A$ and generalize them to a larger set consisting of polytopes $N_{h}$ associated to vectors $h\in\Z^{n}$ as well as $\widehat{\mathcal{P}}$ consisting of polytope functions $ρ_{h}$ corresponding to $N_{h}$. The main contribution contains that (i) obtaining a {\em recurrence construction} of the Laurent expression of a cluster variable in a cluster from its $g$-vector; (ii) proving the subset $\mathcal{P}$ of $\widehat{\mathcal{P}}$ consisting of Laurent polynomials in $\widehat{\mathcal{P}}$ is a strongly positive $\Z Trop(Y)$-basis for $\mathcal{U}(\A)$ consisting of certain universally indecomposable Laurent polynomials when $\A$ is a cluster algebra with principal coefficients. For a cluster algebra $\mathcal A$ over arbitrary semifield $\mathbb P$ in general, $\mathcal{P}$ is a strongly positive $\Z¶$-basis for the intermediate cluster subalgebra $\mathcal{I_P(A)}$ of $\mathcal{U(A)}$. We call $\mathcal P$ the {\em polytope basis}; (iii) constructing some explicit maps among corresponding $F$-polynomials, $g$-vectors, $d$-vectors and cluster variables to characterize their relationship. Moreover, we give three applications of (i), (ii) and (iii) respectively.
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