论文标题
列举和平面组合的单分辨率
Enumerative and planar combinatorics of trivariate monomial resolutions
论文作者
论文摘要
规范的Sylvan分辨率是在最小的多项式环上的任意单元理想的分辨率，该分辨率是最小的，并且具有显式的组合公式。差异是链条链接栅栏权重的晶格路径上的加权总和，它们是通过跨越树的高维类似物相互关联的面部序列。沿着三变量的情况下的晶格路径，可以将这些权重凝结成一个单个权重，以促进组合公式的差速器，以绕过任何计算链条链接围栏的计算。本文中的主要结果表达了三个变量中单元理想的Sylvan矩阵条目，这是一个仅取决于沿相应晶格路径的特定特定koszul简单复合物的数量的晶格路径的总和。某些条目的分子量等于$ \ mathbb {n}^2 $中遵循特定限制的晶格路径的数量。
The canonical sylvan resolution is a resolution of an arbitrary monomial ideal over a polynomial ring that is minimal and has an explicit combinatorial formula for the differential. The differential is a weighted sum over lattice paths of weights of chain-link fences, which are sequences of faces that are linked to each other via higher-dimensional analogues of spanning trees. Along a lattice path in the three-variable case, these weights can be condensed to a single weight contributing to the combinatorial formula for the differential that bypasses any computation of chain-link fences. The main results in this paper express the sylvan matrix entries for monomial ideals in three variables as a sum over lattice paths of simpler weights that depend only on the number of specific Koszul simplicial complexes that lie along the corresponding lattice path. Certain entries have numerators equal to the number of lattice paths in $\mathbb{N}^2$ that follow specific restrictions.
