发表机构
Kerala School of Mathematics（喀拉拉数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文求解了格上移位算子劳伦特环的不定求和问题，其解涉及可估大小的矩阵乘法，等价于计算特定群上同调，且结果可推广到函数环的对应问题。
AI 中文摘要
本文求解了差环$(A, \alpha)$的不定求和问题（ISP），其中$A$是格$\mathbb{Z}^n$上移位算子的劳伦特环，$\alpha$是$A$的任意有限阶环自同构。该解转化为涉及矩阵乘法的有限步骤，矩阵大小可估计，由此可确定解的算术复杂度。这些结果可推广到$\mathbb{Z}^n$上函数环的ISP求解，其中$\alpha$通过对偶作用于该环。本文指出，ISP的求解等价于计算群上同调$H^i([\alpha], A)\\ (i = 0, 1)$，其中$[\alpha]$是由$\alpha$生成的循环群。
英文摘要
This article solves the Indefinite Summation Problem (ISP) for the difference ring $(A, α)$, where $A$ is the Laurent ring of shift operators on the lattice $\Z^n$, and $α$ is any ring automorphism of $A$ of finite order. The solution translates to a finite procedure involving a matrix multiplication, where the size of the matrix can be estimated. It follows that the arithmetic complexity of the solution can also be determined. These results extend to a solution of the ISP for the ring of functions on $\Z^n$, on which $α$ acts by duality. The article points out that the solution to the ISP amounts to calculating the group cohomologies $H^i([α], A), i = 0, 1$, where $[α]$ is the cyclic group generated by $α$.
