# A characterization of generalized exponential polynomials in terms of decomposable functions

## TL;DR

This paper characterizes generalized exponential polynomials on topological commutative semigroups by their decomposability properties of sums, providing a new criterion for identifying such functions.

## Contribution

It establishes a necessary and sufficient condition for a continuous function to be a generalized exponential polynomial based on decomposability of its sum over multiple variables.

## Key findings

- Characterization of generalized exponential polynomials via decomposability
- Equivalent condition involving sum functions on semigroups
- Provides a new perspective for identifying exponential polynomials

## Abstract

Let $G$ be a topological commutative semigroup with unit. We prove that a continuous function $f\colon G\to \cc$ is a generalized exponential polynomial if and only if there is an $n\ge 2$ such that $f(x_1 +\ldots +x_n )$ is decomposable; that is, if $f(x_1 +\ldots +x_n )=\sumik u_i \cd v_i$, where the function $u_i$ only depends on the variables belonging to a set $\emp \ne E_i \subsetneq \{ x_1 \stb x_n \}$, and $v_i$ only depends on the variables belonging to $\{ x_1 \stb x_n \} \se E_i$ $(i=1\stb k)$.