Totally positive elements with $m$ partitions exist in almost all real quadratic fields

TL;DR

This paper investigates the existence and properties of totally positive elements with a fixed number of partitions in real quadratic fields, providing bounds, characterizations, and a comprehensive search for such elements up to seven partitions.

Contribution

It establishes that elements with a given number of partitions exist in almost all real quadratic fields and provides bounds, characterizations, and a complete search for fields with up to seven partitions.

Findings

Existence of $m$-partition elements in almost all real quadratic fields.

Upper bounds for the norm of elements with limited indecomposable representations.

Complete characterization of elements with exactly two representations.

Abstract

In this paper, we study partitions of totally positive integral elements $\alpha$ in a real quadratic field $K$. We prove that for a fixed integer $m \geq 1$, an element with $m$ partition exists in almost all $K$. We also obtain an upper bound for the norm of $\alpha$ that can be represented as a sum of indecomposables in at most $m$ ways, completely characterize the $\alpha$'s represented in exactly $2$ ways, and subsequently apply this result to complete the search for fields containing an element with $m$ partitions for $1 \leq m \leq 7$.