Minimal Mahler Measure in Quartic Galois Number Fields

TL;DR

The paper investigates how the minimal integral Mahler measure $M(\mathcal{O}_K)$ of Galois quartic fields $K$ scales with the discriminant $D_K$, distinguishing biquadratic and cyclic, as well as real versus imaginary cases. It combines Liouville-type Diophantine bounds with Granville's square-free values (the latter conditional on the ABC conjecture) to construct infinite families of fields achieving $M(\mathcal{O}_K) \asymp D_K^{p/q}$ for a broad range of exponents $p/q$, supplemented by unconditional examples. The authors prove improved lower bounds for several families (notably exponent 1/4 for certain imaginary biquadratics and exponent 1/6 to 1/2 for real cyclic/quartic cases) and demonstrate density-type results asserting that many exponents are realized by infinitely many fields under ABC, with extensive treatment of roots of unity and CM-structure. They also provide experimental data for real cyclic quartics up to discriminants of $2\cdot 10^7$, supporting the theoretical bounds and highlighting practical aspects of finding small-height generators.

Abstract

We explore the dependence of the minimal integral Mahler measure of Galois quartic fields on the discriminant of the field. We obtain density results which are conditional on the ABC conjecture as well as several unconditional results.

Figures (2)

• Figure 1: $M(\mathcal{O}_K)$ for real cyclic quartic fields $K$ with $D_K \leq 2\cdot 10^7$
• Figure 2: For real cyclic quartic fields $K$ with $|D_K| \leq 2\cdot 10^7$, the figures show $M(\mathcal{O}_K)(D_K)^{-\frac{1}{4}}$, and $M(\mathcal{O}_K)(D_K)^{-\frac{1}{6}}$, respectively.

Theorems & Definitions (61)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5: Liouville
• Theorem 1.6: Granville
• Proposition 2.1
• Proposition 3.1
• proof
• Corollary 3.2