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

TL;DR

This paper extends previous work on functional equations over compact Abelian groups by showing solutions can be decomposed into simpler parts or structured as step polynomials, enhancing understanding of their algebraic and analytical properties.

Contribution

It introduces a refined class of almost modest P-modules that captures step-polynomial solutions, demonstrating their closure under key operations.

Findings

Solutions decompose into simpler systems or step-polynomial structures.

Step-polynomial solutions form a closed subclass under module operations.

Provides a framework for analyzing solutions to difference and zero-sum equations.

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.