*(English)*Zbl 1221.11082

The central result of this paper is that ${\text{SL}}_{2}\left(\mathbb{Z}\right)$ is a polynomial family, more precisely, there exist polynomials $A$, $B$, $C$, $D$ in 26 variables and with coefficients from $\mathbb{Z}$ such that all integer solutions of

are precisely all quadruples $(A,B,C,D)$ where the variables run through the integers. This solves a problem posed by F. Beukers and going back to T. Skolem.

A series of corollaries and examples employ this theorem to deduce that also other subsets of ${\mathbb{Z}}^{n}$ are polynomial families, e.g., the set of all primitive solutions of any system of linear equations over $\mathbb{Z}$, the set of solutions of several quadratic Diophantine equations, every principal congruence subgroup of ${\text{SL}}_{2}\left(\mathbb{Z}\right)$, the group of regular elements of ${M}_{n\times n}\left(\mathbb{Z}\right)$ and some of its subsets, as ${\text{SL}}_{n}\left(\mathbb{Z}\right)$, ${\text{Spin}}_{n}\left(\mathbb{Z}\right)$, ....

To prove the main result, the author shows in an ingenious way that every $\alpha \in {\text{SL}}_{2}\left(\mathbb{Z}\right)$ can be represented as a product of 9 matrices belonging to several subsets of ${\text{SL}}_{2}\left(\mathbb{Z}\right)$, each of which is a polynomial family (see Prop. 1.5 on p. 993).