A uniform lower bound for the degree-scaled L2 height
Year: 2026
Type: preprint
Abstract: For a nonzero algebraic number alpha of degree d, let h_2(alpha) be its normalized all-place quadratic logarithmic height and put m_2(alpha) = d h_2(alpha). This preprint gives a proof of the bound m_2(alpha) >= sqrt(log(2)/5) when alpha is not a root of unity. The reciprocal-unit argument combines a modulo-8 power congruence, integral Pade approximants and quadratic divisibility of rectangular Hankel minors with a trace-plane logarithmic-capacity estimate. A twisted degree descent removes square-radical exceptions; nonreciprocal units and nonunits are treated separately. The result addresses the p=2 degree-scaled height formulation of Fili and Miner. The same constant holds for all p >= 2. A reciprocal algebraic integer of degree d has house at least exp(sqrt(log(2)/5)/d), without excluding unit-circle conjugates. The paper does not prove classical Lehmer’s conjecture. Version 2 incorporates comments on an internal Version 1. The research and writing were substantially AI-assisted. This is an unrefereed preprint: no complete expert human verification or proof-assistant formalization is documented. LaTeX sources and reproducible finite checks accompany the PDF.
Author: shou
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225786333
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