Computability of a Finite Defining Set for the Algebraic Closure of an Orthogonal Semigroup

Ascertain whether there exists an algorithm that, given orthogonal matrices A_1, \ldots, A_d \in \mathrm{O}_s(\mathbb{Q}) and the recursively enumerable sequence of rational polynomials (p_k) that define the compact algebraic group \mathcal{G} = \overline{\langle A_1, \ldots, A_d \rangle} inside \mathrm{Mat}_s(\mathbb{R}), computes an integer n such that \mathcal{G} = \mathcal{V}(p_1, \ldots, p_n).

Background

For orthogonal generators with rational entries, the closure of the generated semigroup is a compact algebraic group that can be defined as the zero set of a recursively enumerable family of rational polynomials.

By Noetherianity, a finite subfamily suffices, but the authors remark that the number of polynomials needed can be arbitrarily large and it is not known whether this number is algorithmically computable from the input generators.