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

2024/08/15 by Dasgupta, Samit, Kakde, Mahesh
#11J81 #11R27 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2408.08178

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.

Related

• Questions about determinants and polynomials
• The Log-Rank Conjecture: New Equivalent Formulations
• On matrix rank function over bounded arithmetics
• Sign patterns of rational matrices with large rank
• Determinants from homomorphisms