The Indefinite Summation Problem for the Laurent Ring

Abstract

This article solves the Indefinite Summation Problem (ISP) for the difference ring $(A,\alpha )$, where $A$ is the Laurent ring of shift operators on the lattice ${\mathbb{Z}}^n$, and $\alpha$ is any ring automorphism of $A$ of finite order. The space of summable elements in $(A,\alpha )$ is infinite dimensional, and it is possible to write down an explicit basis for it. 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 ${\mathbb{Z}}^n$, on which $\alpha$ acts by duality. The article also points out that the solution to the ISP amounts to calculating the group cohomologies $H^i([\alpha ],A),i=0,1$, where $[\alpha ]$ is the cyclic group generated by $\alpha$.