Hilbert’s Tenth Problem and Elliptic Curves

Coloquio Postgrado, 31.5.2010 a las 17:30

Claudia Degroote
University of Ghent, Belgium

Hilbert’s Tenth Problem and Elliptic Curves

Hilbert’s tenth problem asks for an algorithm that solves the following question: given a polynomial with integer coefficients in any number of unknowns, does this polynomial have an integer zero? In 1970 Matiyasevich proved, building on earlier work of Davis, Putnam and Robinson, that recursively enumerable sets are Diophantine (called the DPRM theorem). From this it followed that there is no algorithm to decide if a polynomial over the integers has integer zeroes, so undecidability for the ring of integers Z has been proved. Many authors have generalized this undecidability problem for other rings and fields, and elliptic curves play an important role in many of the proofs. In 1978, Denef was the first to use an elliptic curve in his proof of the undecidability of R(t). In this talk, I will give the complete proof of this.