Partitions into powers of an algebraic number

TL;DR

This paper investigates how complex numbers can be expressed as sums of powers of an algebraic number, revealing conditions for finiteness of partitions and the universality of the partition function.

Contribution

It characterizes when the partition count is finite for real quadratic algebraic numbers and shows conditions under which the partition function attains all non-negative integers.

Findings

Partition count is finite iff a conjugate of $eta$ exceeds 1 for real quadratic $eta$

Partition function can represent all non-negative integers under certain conditions

Provides new insights into algebraic number-based partitions

Abstract

We study partitions of complex numbers as sums of non-negative powers of a fixed algebraic number $\beta$. We prove that if $\beta$ is real quadratic, then the number of partitions is always finite if and only if some conjugate of $\beta$ is larger than 1. Further, we show that for $\beta$ satisfying a certain condition, the partition function attains all non-negative integers as values.