Abstracts of talks

Diophantine encoding and generalized Cantor's polynomials

Maxim Vsemirrov (Steklov Institute of Mathematics at St. Petersburg)

There are two especilally nice functions among all one-to-one mappings from NxN onto N. These functions are very "simple" and natural as they are given by polynomials. They were already known to G.Cantor. The construction can be generalized to higher dimensions in several ways.

The problem of classification of bijective polynomial mappings from NxNx...xN onto N was raised by C.Smorynski. It gives us another example of a number-theoretic problem which is easy to state and hard to solve. Even in the simplest case of quadratic polynomials in two variables the first proof required very deep tools. Only recently new and more elementary approaches were found. They can be also extended to cubic polynomials in three variables. In that case some properties of elliptic curves come into play.

In my talk I shall give an overview of the problem and concentrate on recent developments.