Heights, regulators and Schinzel’s determinant inequality

Abstract

We prove inequalities that compare the size of an S-regulator with a product of heights of multiplicatively independent S-units. Our upper bound for the S-regulator follows from a general upper bound for the determinant of a real matrix proved by Schinzel. The lower bound for the S-regulator follows from Minkowski's theorem on successive minima and a volume formula proved by Meyer and Pajor. We establish similar upper bounds for the relative regulator of an extension l/k of number fields.2000 Mathematics Subject Classification. 11J25, 11R04, 46B04.

Independent relative units of low height

“…In Section 2 we prove a general identity that connects the volumes of certain star-bodies in Euclidean spaces. In Section 3 we define a generalization of the Schinzel norm that was used in [1], and we record the identity (3.16) for the volume of the unit ball attached to the generalized Schinzel norm. The reason for using the generalized Schinzel norm when working with relative units can be seen in the basic identity (3.22) proved in Lemma 3.1.…”

“…In [1,Theorem 1.2] we proved the existence of a maximal collection of independent S-units such that the product of their heights was bounded by a multiple of the S-regulator. Here we establish an analogous result that bounds the product of the heights of a maximal collection of independent relative units by a multiple of the relative regulator.…”

Lower Bounds for Regulators of Number Fields in terms of their Discriminants

“…In order to sharpen the values of c d and γ d in the inequalities (1.3) that are given in (1.4) and (1.5), we use our results in [1,2] that bound the regulators and relative regulators of an extension of number fields by heights of units and relative units in the number field extension. First we recall that ρ(k) = r(k) if and only if k is a CM-field (see [13, Corollary 1 to Proposition 3.20]).…”

Minkowski’s theorem on independent conjugate units

“…In this work we will only have occasion to use the height h on the subgroup F l ⊆ G l . Further properties of the Weil height on groups are discussed in [1], [3], and [9]. It follows from Minkowski's work in [6] that if F l has positive rank r = r(l), then there exists a coset representative β in F l such that the multiplicative subgroup (1.5) B = σ(β) : σ ∈ Aut(l/Q) ⊆ F l generated by the orbit of β also has rank r. That is, β is a Minkowski unit in F l , and therefore the index [F l : B] is finite.…”

“…In this work we will only have occasion to use the height h on the subgroup F l ⊆ G l . Further properties of the Weil height on groups are discussed in [1], [3], and [9]. It follows from Minkowski's work in [6] that if F l has positive rank r = r(l), then there exists a coset representative β in F l such that the multiplicative subgroup…”