The Indefinite Summation Problem for the Laurent Ring

Abstract

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 $α$.