A bound for the exterior product of \(S\)-units (Q6610067)

The paper under review is a continuation of previous work from the authors that they began in [\textit{S. Akhtari} and \textit{J. D. Vaaler}, Acta Arith. 172, No. 3, 285--298 (2016; Zbl 1346.11041)], where they prove inequalities comparing the size of an \(S\)-regulator with the product of heights of a maximal collection of independent \(S\)-units, and continued in [Acta Arith. 202, No. 4, 389--401 (2022; Zbl 1494.11057)] concerning multiplicative groups of relative units. In the present paper, the authors prove analogous inequalities for the exterior product of a collection of independent \(S\)-units that is not a maximal collection.\N\NMore precisely, let \(k\) be an algebraic number field, \(S\) a finite set of \(r+1\) places of \(k\) containing all the archimedean places, and \(O_S^{\times}\) its multiplicative group of \(S\)-units.\N\NHere is their first result (Theorem 1.1): Given \(1\leq q \leq r\), let \(\alpha_1\), \(\alpha_2\),\(\cdots\), \(\alpha_q\) be multiplicatively independent points in \(O_S^{\times}/ Tor(O_S^{\times})\), and \(\pmb{\alpha}_j = (d_v \log \|\alpha_j\|_v )_{v\in S}\) be their logarithmic embeddings into \( \mathbb{R}^{r +1}\). Then we have: \[\|\pmb{\alpha}_1 \wedge \pmb{\alpha}_2 \wedge \cdots \wedge \pmb{\alpha}_q \|_1 \leq 2^{-q} C(q, r )\prod_{j=1}^q \|\pmb{\alpha}_j\|_1=2^{-q} C(q, r )\prod_{j=1}^q ([k : \mathbb{Q} ]h(\alpha_j)), \] where \(C(r,q)\) is a constant explicitely given, \(\|.\|_1\) is the \(l^1\)-norm \(\| (x)_v\|_1 = \sum_{v\in S} |x_v |\) and its induced map in the exterior algebra \(Ext(\mathbb{R}^{r+1})\), and \(h\) is the classic Weil height.\N\NWith the same notations, a second result is established (Theorem 1.2): There exists \(\beta_j\) (\(1\leq j \leq q\)) such that : \[2^{q} \prod_{j=1}^q ([k : \mathbb{Q} ]h(\beta_j))=\prod_{j=1}^q \|\pmb{\beta}_j\|_1 \leq q! \|\pmb{\alpha}_1 \wedge \pmb{\alpha}_2 \wedge \cdots \wedge \pmb{\alpha}_q \|_1,\] where the logarithm embeddings \(\pmb{\beta}_j\) form a set of generators of a subgroup of the group generated by the \(\pmb{\alpha}_j\).\N\NIn part 2 of the paper, the latter result is linked to a conjecture of F. Rodriguez Villegas on the covolumes of sublattices of the lattice of units of a number field via the logarithm embedding -- a reformulation of Lehmer's conjecture -- stated in [\textit{T. Chinburg} et al., Pac. J. Math. 321, No. 1, 119--165 (2022; Zbl 1521.11065), Appendix], and also discussed in [\textit{F. Amoroso} and \textit{S. David}, Ann. Math. Qué. 45, No. 1, 1--18 (2021; Zbl 1470.11183)].\N\NIn order to establish their two theorems, the authors generalize an inequality for the determinant of a real matrix proved by \textit{A. Schinzel} [Colloq. Math. 38, 319--321 (1978; Zbl 0384.15004)], to more general exterior products of vectors in Euclidean space, in particular using combinatorial methods.