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Ranks of matrices of logarithms of algebraic numbers, I: The theorems of Baker and Waldschmidt–Masser

Samit Dasgupta

Vol. 2 (2023), No. 1, 93–138
Abstract

Let denote the -vector space of logarithms of algebraic numbers. In this expository work, we provide an introduction to the study of ranks of matrices with entries in . We begin by considering a slightly different question; namely, we present a proof of a weak form of Baker’s theorem. This states that a collection of elements of that is linearly independent over is in fact linear independent over ¯. Next we recall Schanuel’s conjecture and prove Ax’s analogue of it over ((t)).

We then consider arbitrary matrices with entries in and state the structural rank conjecture, concerning the rank of a general matrix with entries in . We prove the theorem of Waldschmidt and Masser, which provides a lower bound, giving a partial result toward the structural rank conjecture. We conclude by stating a new conjecture that we call the matrix coefficient conjecture, which gives a necessary condition for a square matrix with entries in to be singular.

Keywords
transcendence theory, Baker's theorem, Ax's theorem, Waldschmidt's theorem, Masser's theorem
Mathematical Subject Classification
Primary: 11J81
Milestones
Received: 2 March 2023
Revised: 12 December 2023
Accepted: 13 December 2023
Published: 31 December 2023
Authors
Samit Dasgupta
Department of Mathematics
Duke University
Durham, NC
United States