The set of all integer solutions of ad-bc=1 is a polynomial family

Number theory 1 Minute

You need to know: Polynomials in k variables, notation {\mathbb Z}^n for the set of vectors x=(x_1, x_2, \dots, x_n) with integer x_i.

Background: We say that a set A \subset {\mathbb Z}^n is a polynomial family with k parameters if there are exist n polynomials P_1, P_2, \dots, P_n in k variables with integer coefficients such that x=(x_1, x_2, \dots, x_n) belongs to A if and only if there exists integers y_1, y_2, \dots, y_k such that x_i = P_i(y_1, y_2, \dots, y_k), \, i=1,2,\dots,n.

The Theorem: On 16th January 2006, Leonid Vaserstein submitted to the Annals of Mathematics a paper in which he proved that the set of all integer solutions of equation x_1x_4-x_2x_3=1 is a polynomial family with 46 parameters.

Short context: What do you mean by “solving” an equation if it has infinitely many solutions? We cannot list all solutions one by one, so the best we can hope for is to present some formulas with parameters which represent all solutions. The Theorem achieves this for the equation x_1x_4-x_2x_3=1, solving a problem which goes back to 1938 paper of Skolem. It immediately implies the existence of polynomial families describing the solutions of some other, more complicated equations.

Links: The original paper is available here. See also Section 10.4 of this book for an accessible description of the Theorem.

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Published by Bogdan Grechuk

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