Back to Algebraic Geometry Seminar: Fall 2008
Algebraic Geometry Seminar
Date: Friday, Nov 21, 2008
Time: 3:30PM - 4:30PM
Location: LCB 222
See also: Comm. Algebra Seminar
Maurice Rojas
Texas A&M University
Title |
Hilbert's Tenth Problem in Low Dimensions |
Abstract |
Hilbert’s Tenth Problem (HTP) asks for an algorithm to decide the existence of integer solutions to arbitrary polynomial equations. HTP was solved in the negative by Davis, Putnam, Robinson, and Matiyasevich around 1970 and, about two decades later, Z. W. Sun proved that undecidability starts already with polynomials in 11 variables. However, while it is a simple matter to find all integer solutions for polynomials in 1 variable, the minimal number of variables where undecidability starts remains a mystery. Furthermore, effective bounds for the size of integer points on curves (when there are only finitely many) also remain unknown in complete generality. We prove a result relating integer points on curves and 3-folds that provides evidence for undecidability starting at 3 variables. We then conclude with a refined result for a p-adic analogue of HTP in 1 variable. The latter result depends subtly on the distribution of primes in arithmetic progressions. We assume no background in number theory. |
Algebraic Geometry