8 problems

Let $M$ be an $m \times n$ matrix with coefficients in $\mathscr{L}$ (or $\mathscr{L}_p$). Write … where $M_i \in M_{m \times n}(\mathbf{Q})$ and the $\lambda_i$ are elements of…

Three-logarithm conjecture. One has

Reciprocal-logarithm conjecture. The numbers

Let $\Omega$ be a lattice in $\mathbb{C}$ with algebraic invariants $g_2,g_3$. Let $s,m,n$ be nonnegative integers with $0\leq m\leq n$. Let $t_1,\dots,t_s$ be $\mathbb{Q}$-linearl…

$p$-adic Structural Rank Conjecture. For every

Four Exponentials Conjecture. If $\det(M)=0$, then the rows or the columns of $M$ are linearly dependent over $\mathbf Q$.

Sum of squared logarithms inequality. If

Let $\mathcal{L}$ be the $\mathbb Q$-vector space of logarithms of algebraic numbers: … For $n\geq 2$, let ${\sf LD}_n^{\mathbb Q}$ denote the relation of $n$ elements of…