The central result of this paper is that is a polynomial family, more precisely, there exist polynomials , , , in 26 variables and with coefficients from such that all integer solutions of
are precisely all quadruples 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 are polynomial families, e.g., the set of all primitive solutions of any system of linear equations over , the set of solutions of several quadratic Diophantine equations, every principal congruence subgroup of , the group of regular elements of and some of its subsets, as , , ....
To prove the main result, the author shows in an ingenious way that every can be represented as a product of 9 matrices belonging to several subsets of , each of which is a polynomial family (see Prop. 1.5 on p. 993).