Ranks of Matrices of Logarithms of Algebraic Numbers II: The Matrix Coefficient Conjecture

TL;DR

This paper introduces a new conjecture in number theory relating to the determinants of matrices of logarithms of algebraic numbers, suggesting conditions under which these determinants vanish after certain transformations.

Contribution

It proposes the Matrix Coefficient Conjecture, offering a novel perspective on the structure of matrices of logarithms of algebraic numbers and their determinants.

Findings

Formulation of the Matrix Coefficient Conjecture

Insight into the structure of matrices with vanishing determinants

Potential implications for nonvanishing determinant problems in number theory

Abstract

Many questions in number theory concern the nonvanishing of determinants of square matrices of logarithms (complex or p-adic) of algebraic numbers. We present a new conjecture that states that if such a matrix has vanishing determinant, then after a rational change of basis on the left and right, it can be made to have a vanishing coefficient.