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

Abstract

Ranks of matrices of logarithms of algebraic numbers, IThe theorems of Baker and Waldschmidt-Masser Samit DasguptaLet 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. 1. Introduction 93 2. Baker's theorem 96 3. Ax's theorem 105 4. The structural rank conjecture 111 5. The theorem of Waldschmidt and Masser 121 6. The matrix coefficient conjecture 136 Acknowledgements 137 References 137 MSC2020: 11J81.