Large Galois groups with applications to Zariski density

Abstract

We introduce the first provably efficient algorithm to check if a finitely generated subgroup of an almost simple semi-simple group over the rationals is Zariski-dense. We reduce this question to one of computing Galois groups, and to this end we describe efficient algorithms to check if the Galois group of a polynomial $p$ with integer coefficients is "generic" (which, for arbitrary polynomials of degree $n$ means the full symmetric group $S_n,$ while for reciprocal polynomials of degree $2n$ it means the hyperoctahedral group $C_2 \wr S_n.$). We give efficient algorithms to verify that a polynomial has Galois group $S_n,$ and that a reciprocal polynomial has Galois group $C_2 \wr S_n.$ We show how these algorithms give efficient algorithms to check if a set of matrices $\mathcal{G}$ in $\mathop{SL}(n, \mathbb{Z})$ or $\mathop{Sp}(2n, \mathbb{Z})$ generate a \emph{Zariski dense} subgroup. The complexity of doing this in$\mathop{SL}(n, \mathbb{Z})$ is of order $O(n^4 \log n \log \|\mathcal{G}\|)\log \epsilon$ and in $\mathop{Sp}(2n, \mathbb{Z})$ the complexity is of order $O(n^8 \log n\log \|\mathcal{G}\|)\log \epsilon$ In general semisimple groups we show that Zariski density can be confirmed or denied in time of order $O(n^14 \log \|\mathcal{G}\|\log \epsilon),$ where $\epsilon$ is the probability of a wrong "NO" answer, while $\|\mathcal{G}\|$ is the measure of complexity of the input (the maximum of the Frobenius norms of the generating matrices). The algorithms work essentially without change over algebraic number fields, and in other semi-simple groups. However, we restrict to the case of the special linear and symplectic groups and rational coefficients in the interest of clarity.