Heights, regulators and Schinzel's determinant inequality (Q2787087)

Let \(k\) be an algebraic number field, \(S\) be a finite set of places of \(k\) containing the infinite ones, with \(r:= (\text{card\,}S)- 1\) and \(r\geq 1\), \(W\) be the (finite) subgroup of the roots of unity of \(k^*\), \(O_S\) be the ring of \(S\)-integers of \(k\), and \({\mathfrak D}_S:= O^*_S/W\). Let NEWLINE\[NEWLINE\{\alpha_j\mid 1\leq j\leq r\}\subseteq{\mathfrak D}_S\quad\text{and}\quad A:= \Biggl\{\prod^r_{j=1} \alpha^{n_j}_j\mid n\in\mathbb Z^r\Biggr\}.NEWLINE\]NEWLINE Suppose that \({\mathfrak D}_S/A\) is a finite group; the authors prove then that NEWLINE\[NEWLINER_S\text{\,card}({\mathfrak D}_S/A)\leq \prod^r_{j=1} ([k: \mathbb Q] h(\alpha_j)),NEWLINE\]NEWLINE where \(R_S\) is the \(S\)-regulator of \(k\) and \(h: k^*\to\mathbb R_+\) is the logarithmic Weil height. The authors obtain also an analogue of this result for a finite field extension \(K\mid k\), making use of the notion of the relative regulator of such an extension. In the opposite direction, they prove that any subgroup \(B\) of finite index in \({\mathfrak D}_S\) can be generated by the elements \(\beta_1,\dots,\beta_r\) such that NEWLINE\[NEWLINE\prod^r_{j=1} ([k:\mathbb Q] h(\beta_j))\leq c\,R_S\text{\,card}({\mathfrak D}_S/B),NEWLINE\]NEWLINE with \(c(r):= 2^r(r!)^3((2r)!)^{-1}\). The authors give several applications of their results, in particular, to a form of Lehmer's problem (discussed, for example, at some length by \textit{C. Smyth} [Number theory and polynomials. Lond. Math. Soc. Lect. Note Ser. 352, 322--349 (2008; Zbl 1334.11081)].