On a problem of A. M. Odlyzko on algebraic units of bounded degree (Q1912644)

Let \(K\) be an algebraic number field and denote by \(M(K)\) the maximal length \(m\) of a sequence \((\varepsilon_1, \dots, \varepsilon_m)\) in \(K\) such that \(\varepsilon_i - \varepsilon_j\) is a unit for all \(i<j\) with \(1\leq i\), \(j\leq m\). According to a theorem of \textit{H. W. Lenstra jun.} [Invent. Math. 38, 237-254 (1977; Zbl 0341.12004)], if \(M(K)\) is large enough (relative to some parameters of \(K)\), then \(K\) is a norm-Euclidean field. Denote by \(M(n)\) the maximal number \(m\) of algebraic units \((\varepsilon_1, \dots, \varepsilon_m)\) of degree at most \(n\) (contained possibly in different fields) such that \( \varepsilon_i - \varepsilon_j\) is a unit for all \(i<j\) with \(1\leq i\), \(j\leq m\). Answering a question of Odlyzko, in the present paper the author proves \[ M(n) < \exp \exp \bigl\{39n (n^{2n+1})! \bigr\} \] by using a result of his on the number of solutions of decomposable form equations.