Ranks of matrices of logarithms of algebraic numbers, I : The theorems of Baker and Waldschmidt–Masser
Samit Dasgupta · Essential Number Theory · 2023
Let L denote the -ޑvector space of logarithms of algebraic numbers.In this expository work, we provide an introduction to the study of ranks of matrices with entries in L .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 L that is linearly independent over ޑ is in fact linear independent over .ޑ Next we recall Schanuel's conjecture and prove Ax's analogue of it over ((ރt)).We then consider arbitrary matrices with entries in L and state the structural rank conjecture, concerning the rank of a general matrix with entries in L .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 L to be singular.