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Bertrand's and Rodriguez Villegas' Conjecture for real quartic Galois extensions of the rationals

Dohyeong Kim, Seungho Song

TL;DR

The paper proves an improved universal lower bound for the Bertrand-Rodriguez Villegas constant in the exterior square ($j=2$) of the logarithmic unit embedding for real quartic Galois extensions by treating separately the Klein four-group and cyclic Galois types. It leverages Pohst-type inequalities and explicit Galois-module structures of unit groups, including biquadratic and relative-unit decompositions, to bound the 1-norm of nonzero wedge elements from below. The main result is $A_{4,2} \ge 2\sqrt{6}\,\log^2\left(\frac{1+\sqrt{5}}{2}\right) \approx 1.134$, strengthening the previous bound $A_{4,2} > 2\sqrt{3}\,\log^2\left(\frac{1+\sqrt{5}}{2}\right) \approx 0.802$. The approach clarifies the role of the Galois group in the geometry of units and provides concrete estimates that complement connections to Lehmer’s conjecture (for $j=1$) and Zimmert’s regulator bounds (for $j=\operatorname{rank}(\mathcal{O}_L^*)$). Overall, the work advances the evidence for the Bertrand-Rodriguez Villegas conjecture in the real quartic Galois setting and isolates precise, case-based methods for exterior-power norm bounds.

Abstract

The conjecture due to Bertrand and Rodriguez Villegas asserts that the 1-norm of the nonzero element in an exterior power of the units of a number field has a certain lower bound. For the exterior square case of totally real quartic extensions of the rationals, Costa and Friedman gave a lower bound of 0.802. We prove that the bound can be improved to 1.134 when the extension is further assumed to be Galois.

Bertrand's and Rodriguez Villegas' Conjecture for real quartic Galois extensions of the rationals

TL;DR

The paper proves an improved universal lower bound for the Bertrand-Rodriguez Villegas constant in the exterior square () of the logarithmic unit embedding for real quartic Galois extensions by treating separately the Klein four-group and cyclic Galois types. It leverages Pohst-type inequalities and explicit Galois-module structures of unit groups, including biquadratic and relative-unit decompositions, to bound the 1-norm of nonzero wedge elements from below. The main result is , strengthening the previous bound . The approach clarifies the role of the Galois group in the geometry of units and provides concrete estimates that complement connections to Lehmer’s conjecture (for ) and Zimmert’s regulator bounds (for ). Overall, the work advances the evidence for the Bertrand-Rodriguez Villegas conjecture in the real quartic Galois setting and isolates precise, case-based methods for exterior-power norm bounds.

Abstract

The conjecture due to Bertrand and Rodriguez Villegas asserts that the 1-norm of the nonzero element in an exterior power of the units of a number field has a certain lower bound. For the exterior square case of totally real quartic extensions of the rationals, Costa and Friedman gave a lower bound of 0.802. We prove that the bound can be improved to 1.134 when the extension is further assumed to be Galois.
Paper Structure (6 sections, 10 theorems, 76 equations)

This paper contains 6 sections, 10 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. The lower bound due to Costa and Friedman
  3. The Klein four-group Case
  4. The Cyclic Group Case
  5. The Case Q=1
  6. The Case Q=2

Key Result

Theorem 1.2

Let $L$ be a real quartic Galois extension of $\mathbb{Q}$. Then for any nonzero $w \in \bigwedge^2\operatorname{LOG}({\mathcal{O}_L^*})$. In other words,

Theorems & Definitions (17)

  • Definition 1.1
  • Conjecture : Bertrand--Rodriguez Villegas
  • Theorem 1.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Definition 4.1
  • Lemma 4.2
  • Lemma 4.3
  • ...and 7 more