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

Abstract

Let $K$ be a number field. In the terminology of Nagell a unit $\epsilon$ of $K$ is called {\it exceptional} if $1-\epsilon$ is also a unit. The existence of such a unit is equivalent to the fact that the unit equation ${\epsilon }_1+{\epsilon }_2+{\epsilon }_3=0$ is solvable in units ${\epsilon }_1,{\epsilon }_2,{\epsilon }_3$ 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 $k\ge 3$, denoted by $\ell (K)$, such that the unit equation ${\epsilon }_1+\cdots +{\epsilon }_k=0$ is solvable in units ${\epsilon }_1,\dots ,{\epsilon }_k$ 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.

We expand the proof of Theorem 2.4. Although the original proof is correct we feel that it is worth to give more explanation in one of the cases