Ranks of matrices of logarithms of algebraic numbers I: the theorems of Baker and Waldschmidt-Masser

TL;DR

The paper surveys the landscape of ranks of matrices with coefficients in the $\ extbf{Q}$-vector space $\mathscr L$ of logarithms of algebraic numbers, connecting Baker’s theory, Ax’s function-field analogue, and the Structural Rank Conjecture to deep conjectures in transcendence and Iwasawa theory. It develops the machinery around auxiliary polynomials (Baker’s method, Siegel's lemma, and derivative vanishing), proves Ax’s theorem, and elucidates the structural-rank philosophy that Schanuel’s conjecture implies—and, conversely, Roy’s results tie the two directions together. The Waldschmidt–Masser framework provides strong, unconditional evidence toward the Structural Rank Conjecture, with a parallel $p$-adic theory informing Leopoldt’s and Gross–Kuz'min conjectures. The Matrix Coefficient Conjecture encapsulates a refined, testable criterion for singular matrices over $\mathscr L}$, tying together determinant-based, representation-theoretic, and interpolation-determinant techniques and suggesting directions for further progress. Overall, the work builds a cohesive bridge between transcendence theory, Diophantine approximation, and $p$-adic Iwasawa theory through rank phenomena of logarithmic matrices.

Abstract

Let $\mathscr{L}$ denote the $\mathbf{Q}$-vector space of logarithms of algebraic numbers. In this expository work, we provide an introduction to the study of ranks of matrices with coefficients in $\mathscr{L}$. 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 $\mathscr{L}$ that is linearly independent over $\mathbf{Q}$ is in fact linear independent over $\overline{\mathbf{Q}}$. Next we recall Schanuel's Conjecture and prove Ax's analogue of it over $\mathbf{C}((t))$. We then consider arbitrary matrices with coefficients in $\mathscr{L}$ and state the Structural Rank Conjecture, which gives a conjecture for the rank of a general matrix with coefficients in $\mathscr{L}$. 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 coefficients in $\mathscr{L}$ to be singular.

Theorems & Definitions (80)

• Theorem 1.1: Gelfond--Schneider
• Theorem 1.2: Baker, 1966
• Conjecture 1.3: Structural Rank Conjecture
• Conjecture 1.4
• Theorem 1.5: Roy
• Theorem 1.6: Waldschmidt--Masser
• Theorem 2.1
• Theorem 2.2
• proof
• Lemma 2.3