Partial difference equations over compact Abelian groups, II: step-polynomial solutions

Abstract

This paper continues an earlier work on the structure of solutions to two classes of functional equation. Let $Z$ be a compact Abelian group and $U_1$, \ldots, $U_k \leq Z$ be closed subgroups. Given $f:Z\to\mathbb{T}$ and $w \in Z$, one defines the differenced function \[d_wf(z) := f(z+w) - f(z).\] In this notation, we shall study solutions to the system of difference equations \[d_{u_1}\cdots d_{u_k}f \equiv 0 \quad \forall (u_1,\ldots,u_k) \in \prod_{i\leq k}U_i,\] and to the zero-sum problem \[f_1 + \cdots + f_k = 0\] for functions $f_i:Z\to \mathbb{T}$ that are $U_i$-invariant for each $i$. Part I of this work showed that the $Z$-modules of solutions to these problems can be described using a general theory of `almost modest $\P$-modules'. Much of the global structure of these solution $Z$-modules could then be extracted from results about the closure of this general class under certain natural operations, such as forming cohomology groups. The main result of the present paper is that solutions to either problem can always be decomposed into summands which either solve a simpler system of equations, or have some special `step polynomial' structure. This will be proved by augmenting the definition of `almost modest $\mathcal{P}$-modules' further, to isolate a subclass in which elements can be represented by the desired `step polynomials'. We will then find that this subclass is closed under the same operations.