On the smallest number of terms of vanishing sums of units in number fields

TL;DR

This paper investigates the minimal number of units in a number field that sum to zero, extending the concept of exceptional units, providing bounds and properties of this minimal number across various fields, and exploring applications to arithmetic graphs.

Contribution

It introduces and analyzes the invariant , the smallest number of units summing to zero in a number field, offering explicit bounds and properties for fields of degree up to 4 and cyclotomic fields.

Findings

Established explicit upper bounds for  in various number fields.

Proved properties of  including its magnitude and parity.

Connected  to the representability of cycles in arithmetic graphs.

Abstract

Let $K$ be a number field. In the terminology of Nagell a unit $\varepsilon$ of $K$ is called {\it exceptional} if $1-\varepsilon$ is also a unit. The existence of such a unit is equivalent to the fact that the unit equation $\varepsilon_1+\varepsilon_2+\varepsilon_3=0$ is solvable in units $\varepsilon_1,\varepsilon_2,\varepsilon_3$ of $K$. Numerous number fields have exceptional units. They have been investigated by many authors, and they have important applications. In this paper we deal with a generalization of exceptional units. We are interested in the smallest integer $k$ with $k\geq 3$, denoted by $\ell(K)$, such that the unit equation $\varepsilon_1+\dots+\varepsilon_k=0$ is solvable in units $\varepsilon_1,\dots,\varepsilon_k$ of $K$. If no such $k$ exists, we set $\ell(K)=\infty$. Apart from trivial cases when $\ell(K)=\infty$, we give an explicit upper bound for $\ell(K)$. We obtain several results for $\ell(K)$ in number fields of degree at most $4$, cyclotomic fields and general number fields of given degree. We prove various properties of $\ell(K)$, including its magnitude, parity as well as the cardinality of number fields $K$ with given degree and given odd resp. even value $\ell(K)$. Finally, as an application, we deal with certain arithmetic graphs, namely we consider the representability of cycles. We conclude the paper by listing some problems and open questions.