论文标题
简单的彩色多项式作为史丹利的希尔伯特系列 - reisner戒指
Simplicial chromatic polynomials as Hilbert series of Stanley--Reisner rings
论文作者
论文摘要
我们发现了简单复合物的家族，其中cooper-de silva-sazdanovic \ cite {cdss}定义的简单色度多项式是希尔伯特（Hilbert）系列的斯坦利（Stanley） - 辅助简单综合体的Reisner Rings。结果，这种广义的色度评价由辅助简单络合物的$ h $向量确定。除了将相关的结果概括为图和矩形外，所使用的简单复合物还使我们能够考虑不一定是被认为是图形的问题的问题。一些示例包括支持环分多项式的支持，多项式多项式的对数凹度特性以及多项式的某些翻译以及多项式之间的对称关系。 如果涉及的$ h $ - 向量有足够大的条目，那么希尔伯特（Hilbert）系列是一些$ k $ - 代数的希尔伯特（Hilbert）多项式。由于$ h $ - 向量与简单色度多项式之间的连接，我们还发现了简单的复合物，其$ h $ vectors的$ h $ vectors是通过类似于图形的删除收集关系的简单复合物的加法关系来确定的。所使用的构造涉及构型的概括欧拉（Euler）特征的构型特征和图形的多项式。
We find families of simplicial complexes where the simplicial chromatic polynomials defined by Cooper--de Silva--Sazdanovic \cite{CdSS} are Hilbert series of Stanley--Reisner rings of auxiliary simplicial complexes. As a result, such generalized chromatic polynomials are determined by $h$-vectors of auxiliary simplicial complexes. In addition to generalizing related results on graphs and matroids, the simplicial complexes used allow us to consider problems that are not necessarily analogues of those considered for graphs. Some examples include supports of cyclotomic polynomials, log concavity properties of a polynomial or some translate of the polynomial, and symmetry relations between a polynomial and its reciprocal polynomial. If the $h$-vectors involed have sufficiently large entries, the Hilbert series are Hilbert polynomials of some $k$-algebra. As a consequence of connections between $h$-vectors and simplicial chromatic polynomials, we also find simplicial complexes whose $h$-vectors are determined by addition-contraction relations of simplicial complexes analogous to deletion-contraction relations of graphs. The constructions used involve generalizations of relations Euler characteristics of configuration spaces and chromatic polynomials of graphs.
