Abstract

Let $\mathcal{ℒ}$ denote the $\mathbb{Q}$-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 $\mathcal{ℒ}$. 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 $\mathcal{ℒ}$ that is linearly independent over $\mathbb{Q}$ is in fact linear independent over $\overline{\mathbb{Q}}$. Next we recall Schanuel’s conjecture and prove Ax’s analogue of it over $ℂ((t))$.

We then consider arbitrary matrices with entries in $\mathcal{ℒ}$ and state the structural rank conjecture, concerning the rank of a general matrix with entries in $\mathcal{ℒ}$. 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 $\mathcal{ℒ}$ to be singular.

Keywords

Mathematical Subject Classification

Milestones

Authors