Decidability of membership problems for flat rational subsets of $\mathrm{GL}(2,\mathbb{Q})$ and singular matrices

Abstract

This work relates numerical problems on matrices over the rationals to symbolic algorithms on words and finite automata. Using exact algebraic algorithms and symbolic computation, we prove various new decidability results for $2\times 2$ matrices over $\mathbb{Q}$. For that, we introduce the concept of flat rational sets: if $M$ is a monoid and $N$ is a submonoid, then ``flat rational sets of $M$ over $N$'' are finite unions of the form $L_0g_1L_1 \cdots g_t L_t$ where all $L_i$'s are rational subsets of $N$ and $g_i\in M$. We give quite general sufficient conditions under which flat rational sets form an effective relative Boolean algebra. As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of $GL(2,\mathbb{Q})$ over $GL(2,\mathbb{Z})$ is decidable (in singly exponential time). It is possible that such a strong decidability result cannot be pushed any further inside $GL(2,\mathbb{Q})$. We also show a dichotomy for nontrivial group extension of $GL(2,\mathbb{Z})$ in $GL(2,\mathbb{Q})$: if $G$ is a f.g. group such that $GL(2,\mathbb{Z}) < G \leq GL(2,\mathbb{Q})$, then either $G\cong GL(2,\mathbb{Z})\times Z^k$, for some $k\geq 1$, or $G$ contains an extension of the Baumslag-Solitar group $BS(1,q)$, with $q\geq 2$, of infinite index. In the first case of the dichotomy the membership problem for $G$ is decidable but the equality problem for rational subsets of $G$ is undecidable. In the second case, decidability of the membership problem for rational subsets in $G$ is open. In the last section we prove new decidability results for flat rational sets that contain singular matrices. In particular, we show that the membership problem is decidable (in doubly exponential time) for flat rational subsets of $Q^{2 \times 2}$ over the submonoid that is generated by the matrices from $Z^{2 \times 2}$ with determinants in $\{-1,0,1\}$