[Paper Review] Orthogonal decomposition of the space of algebraic numbers and Lehmer's problem

Paul Fili, Zachary Miner|ArXiv.org|Nov 10, 2009

Algebraic Geometry and Number Theory7 references3 citations

TL;DR

This paper introduces $L^p$-norms on the space of algebraic numbers modulo torsion using orthogonal decompositions by Galois fields and degrees, linking them to Lehmer's problem. It proves that $L^p$-Lehmer conjectures are equivalent to classical conjectures: the original Lehmer conjecture for $p=1$ and the Schinzel-Zassenhaus conjecture for $p=\infty$, establishing a new framework for studying small Mahler measures via functional analysis on $L^p$ spaces.

ABSTRACT

We introduce vector space norms associated to the Mahler measure by using the L^p norm versions of the Weil height recently introduced by Allcock and Vaaler. In order to do this, we determine orthogonal decompositions of the space of algebraic numbers modulo torsion by Galois field and degree. We formulate L^p Lehmer conjectures involving lower bounds on these norms and prove that these new conjectures are equivalent to their classical counterparts, specifically, the classical Lehmer conjecture in the p = 1 case and the Schinzel-Zassenhaus conjecture in the p = infinity case.

Motivation & Objective

• To reformulate Lehmer's problem using $L^p$-norms on the space of algebraic numbers modulo torsion.
• To construct orthogonal decompositions of this space based on Galois fields and degrees.
• To establish equivalence between $L^p$-Lehmer conjectures and classical conjectures in Diophantine approximation.
• To define and analyze the Mahler $p$-norm and its properties, especially for $p=2$.
• To show that the discrete topology on the unit group $\Gamma$ is equivalent to the $L^p$-Lehmer conjecture, linking number theory to functional analysis.

Proposed method

• Use the Weil height and Mahler measure to define a function space $\mathcal{F}$ of algebraic numbers modulo torsion.
• Apply the $L^p$ norm framework from Allcock and Vaaler to define $L^p$-versions of the Weil height.
• Construct orthogonal decompositions of $\mathcal{F}$ by Galois fields and by degree using projection operators $P_K$ and $T^{(n)}$.
• Define the Mahler $p$-norm as $\|f\|_{m,p} = \|Mf\|_p$, where $Mf = \sum n T^{(n)}f$, linking it to the classical Mahler measure.
• Prove that the $L^p$-Lehmer conjecture holds if and only if the unit group $\Gamma$ is discrete in the $L^p$-topology.
• Use spectral decomposition and norm inequalities to relate $\|f\|_{m,p}$ to $\|T^{(n)}f\|_2$ and $\|P_K f\|_{m,p}$.

Experimental results

Research questions

• RQ1 How can the Mahler measure be reinterpreted as an $L^p$-norm on a function space of algebraic numbers modulo torsion?
• RQ2 What orthogonal decompositions of $\mathcal{F}$ by Galois fields and degrees enable a functional-analytic formulation of Lehmer's problem?
• RQ3 Is the $L^p$-Lehmer conjecture equivalent to the classical Lehmer conjecture for $p=1$ and the Schinzel-Zassenhaus conjecture for $p=\infty$?
• RQ4 Does the Mahler $2$-norm arise from a Hilbert space structure, and what are its spectral properties?
• RQ5 Is the unit group $\Gamma$ discrete in the $L^p$-topology if and only if the $L^p$-Lehmer conjecture holds?

Key findings

• The $L^p$-Lehmer conjecture is equivalent to the classical Lehmer conjecture when $p=1$, and to the Schinzel-Zassenhaus conjecture when $p=\infty$.
• The Mahler $2$-norm is defined as $\|f\|_{m,2} = \left(\sum_{n=1}^\infty n^2 \|T^{(n)}f\|_2^2\right)^{1/2}$, and it arises from an inner product, making the completion of $\mathcal{F}$ a Hilbert space.
• The orthogonal decomposition of $\mathcal{F}$ into subspaces $V_K^{(n)}$ by Galois field $K$ and degree $n$ allows for a spectral analysis of the Mahler norm.
• The projection operator $P_K$ is a norm-one projection in $L^p$, and it commutes with the $T^{(n)}$ operators, preserving the $L^p$-norm structure.
• The unit group $\Gamma$ is discrete in the $L^p$-topology if and only if the $L^p$-Lehmer conjecture holds, providing a topological characterization of the conjecture.
• For $p=2$, the inequality $\|f\|_{m,2} \leq \delta(f) \|f\|_2$ holds, where $\delta(f)$ is the degree of the minimal field of $f$, linking the Mahler norm to the degree.