SEMINARIO DI MATEMATICA

martedì 28 ottobre 2008

ore 16.00

Scuola Normale Superiore

Pisa

(Aula Bianchi)

Jeroen Demeyer

Scuola Normale Superiore, Pisa

Terrà  un seminario dal titolo:

“Undecidability and elliptic curves”

Abstract

Hilbert's Tenth Problem is the following:

find an algorithm which, given a polynomial f(x1, … , xn) in Z[x1, … , xn], tells whether or not it has a zero (x1, … , xn) in Zn.

It was shown in 1970 by Y. Matiyasevich, building on earlier work by M. Davis, H. Putnam and J. Robinson, that such an algorithm does not exist.

In other words: general diophantine equations over the integers are undecidable.

This problem can be generalized by replacing Z by a different ring.

In this talk we will concentrate on fields (usually the hardest case).

For finite fields, algebraically closed fields, R and Qp one has decidability for polynomial equations; in all other cases where the answer is known

(such as R(t) or C(t1, t2), we have undecidability. Perhaps surprisingly, elliptic curves play an important role in many of these undecidability proofs.

In this talk, I will give an overview of the ideas in these proofs and in particular how elliptic curves are used.

Tutti gli interessati sono invitati a partecipare.

                                                                           La Segreteria della

                                                                                              Classe di Scienze