Integral points on varieties defined by matrix factorization into elementary matrices

Abstract

Let ${\mathcal O}$ be the ring of $S$-integers in a number field $K$. For $A\in\rm{SL}_{2}(\mathcal{O})$ and $k\geq 1$, we define matrix-factorization varieties $V_k(A)$ over ${\mathcal O}$ which parametrize factoring $A$ into a product of $k$ elementary matrices; the equations defining $V_k(A)$ are written in terms of Euler's continuant polynomials. We show that the $V_k(A)$ are rational $(k-3)$-folds with an inductive fibration structure. We combine this geometric structure with arithmetic results to study the Zariski closure of the ${\mathcal O}$-points of $V_k(A)$. We prove that for $k\geq 4$ the ${\mathcal O}$-points on $V_k(A)$ are Zariski dense if $V_{k}(A)({\mathcal O})\neq\emptyset$ assuming the group of units ${\mathcal O}^{\times}$ is infinite. This shows that if $A$ can be written as a product of $k\geq 4$ elementary matrices, then this can be done in infinitely many ways in the strongest sense possible. This can then be combined with results on factoring into elementary matrices for ${\rm SL}_{2}({\mathcal O})$. One result is that for $k\geq 9$ the ${\mathcal O}$-points on $V_{k}(A)$ are Zariski dense if ${\mathcal O}^{\times}$ is infinite.