On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices

TL;DR

This paper addresses the computational problem of obtaining the Zariski closure of a finitely generated matrix group and provides new elementary-time bounds. By developing a quantitative structure lemma that isolates a finite-index subgroup H with a commutative quotient over the unipotent part, and by leveraging a pistil construction for semisimple components, the authors bound the degree of the defining polynomials by $$(\log h)^{2^{|S|^{\exp^6(\mathrm{poly}(n))}}}$$ (and a sharper bound when all elements are semisimple). The combination of these degree bounds with algorithmic reductions (including Maser's bounds on eigenvalue multiplicative relations and Dube’s bound on elimination) yields an explicit, elementary-time algorithm to compute the Zariski closure, along with a finite bound on the length of chains of linear algebraic groups over a fixed number field. The work also extends the framework to unitary/orthogonal cases, yielding triply-exponential chain bounds, and discusses extensions to number fields and the dependence on dimension and height. Overall, the results provide the first fully explicit, elementary-time complexity bounds for this fundamental algebraic problem, with implications for applications in quantum computing and program verification where Zariski-closure representations enable powerful algebraic tools.

Abstract

We investigate the complexity of computing the Zariski closure of a finitely generated group of matrices. The Zariski closure was previously shown to be computable by Derksen, Jeandel, and Koiran, but the termination argument for their algorithm appears not to yield any complexity bound. In this paper we follow a different approach and obtain a bound on the degree of the polynomials that define the closure. Our bound shows that the closure can be computed in elementary time. We also obtain upper bounds on the length of chains of linear algebraic groups, where all the groups are generated over a fixed number field.

Figures (3)

• Figure 1: Assembling the arguments of \ref{['lem:hrushovski_alggroup', 'lem:hrushovski_semisimple']}: we take two consecutive quotients to obtain a finite group. We then pull-back to obtain a group $H$ with the desired properties. Note that the dimension increases after each quotient and the lemmata relate $p$ and $q$ to $n$.
• Figure 2: Graphical representation of \ref{['lem:hrushovski_semisimple']}: given a group $G$ that is the closure of a group generated by rational matrices and is comprised of semisimple elements, we define the pistil $P$ as the intersection of the maximal tori containing ${G}^\circ$; note that $P$ is not necessarily a subgroup of $G$.
• Figure 3: Graphical representation of \ref{['lem:hrushovski_alggroup']}: given a group $G=\overline{\left\langle S\right\rangle}^{}$ for $S\subseteq \mathop{\mathrm{GL}}\nolimits_n(\mathbb{Q})$ a set of rational matrices, we identify a normal subgroup $H$ with the desired properties. We construct a homomorphism $\phi_U \colon G \to \mathop{\mathrm{GL}}\nolimits_p(\overline{\mathbb{Q}})$ with kernel $U$, and define $H$ to be the preimage of the pistil $P$ of $\phi_U(G) \cong G/U$ under $\phi_U$.

Theorems & Definitions (45)

• Theorem 1: Degree bound
• Theorem 1: Main result
• Lemma 1: Quantitative Structure Lemma
• Lemma 1: Pistil Lemma
• Theorem 2: Feng
• Lemma 2
• Lemma 2: Pistil Lemma
• proof
• Claim 2.1
• proof : Proof