Let
$\mathcal{\mathcal{L}}$ 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
$\mathcal{\mathcal{L}}$. 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
$\mathcal{\mathcal{L}}$ 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 $\u2102((t))$.
We then consider arbitrary matrices with entries in
$\mathcal{\mathcal{L}}$ and
state the structural rank conjecture, concerning the rank of a general matrix with entries
in
$\mathcal{\mathcal{L}}$.
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
$\mathcal{\mathcal{L}}$ to be
singular.
