论文标题
对称分解和Veronese Construction
Symmetric decompositions and the Veronese construction
论文作者
论文摘要
我们研究序列的有理生成函数$ \ {a_n \} _ {n \ geq 0} $与多项式一致，并调查了分子多项式的对称分解，用于子序列$ \ {a_ {a_ {rn}我们证明，如果按$ \ {a_n \} _ {n \ geq 0} $的numerator多项式为单位$ s $，并且其系数满足一组天然线性不平等，那么计数器的对称分解，那么$ \ \ {a_ iS i i i} $ ceal for $ \ {rn} $ nece n eemen n eemen n eemer $ r \ geq \ max \ {s，d+1-s \} $。此外，如果$ \ {a_n \} _ {n \ geq 0} $的分子多项式是对称的，那么我们表明$ \ {a_ {rn} \} _ {n \ geq 0} $的对称分解是交织的。我们将结果应用于Ehrhart系列的晶格多型。特别是，我们得到的是，每当$ d $二维的晶格$ s $的$ h^\ ast $ - 多物质元素都具有实用的对称分解，只要扩张因子$ r $满足$ r \ r \ geq \ geq \ geq \ geq \ max \ max \ {s，d+1-s s，d+1-s \ \ \} $。此外，如果多层是戈伦斯坦，那么这种分解是交错的。
We study rational generating functions of sequences $\{a_n\}_{n\geq 0}$ that agree with a polynomial and investigate symmetric decompositions of the numerator polynomial for subsequences $\{a_{rn}\}_{n\geq 0}$. We prove that if the numerator polynomial for $\{a_n\}_{n\geq 0}$ is of degree $s$ and its coefficients satisfy a set of natural linear inequalities then the symmetric decomposition of the numerator for $\{a_{rn}\}_{n\geq 0}$ is real-rooted whenever $r\geq \max \{s,d+1-s\}$. Moreover, if the numerator polynomial for $\{a_n\}_{n\geq 0}$ is symmetric then we show that the symmetric decomposition for $\{a_{rn}\}_{n\geq 0}$ is interlacing. We apply our results to Ehrhart series of lattice polytopes. In particular, we obtain that the $h^\ast$-polynomial of every dilation of a $d$-dimensional lattice polytope of degree $s$ has a real-rooted symmetric decomposition whenever the dilation factor $r$ satisfies $r\geq \max \{s,d+1-s\}$. Moreover, if the polytope is Gorenstein then this decomposition is interlacing.
