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Polynomial parametrization for the solutions of Diophantine equations and arithmetic groups. (English) Zbl 1221.11082

The central result of this paper is that SL 2 () is a polynomial family, more precisely, there exist polynomials A, B, C, D in 26 variables and with coefficients from 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 n 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 SL 2 (), the group of regular elements of M n×n () and some of its subsets, as SL n (), Spin n (), ....

To prove the main result, the author shows in an ingenious way that every αSL 2 () can be represented as a product of 9 matrices belonging to several subsets of SL 2 (), each of which is a polynomial family (see Prop. 1.5 on p. 993).

11D09Quadratic and bilinear diophantine equations
11F06Structure of modular groups and generalizations