Hilbert's 10th Problem, Heights, and Search Bounds for Rational Points on Varieties, Parts I and II

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Mathematics (CMC)

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Hilbert's 10-th problem (in its modern formulation) asks if there exists an algorithm that, given a Diophantine equation (or a system of such equations), can decide whether it has a (non-trivial) solution in integers. In a famous work by Matijasevich (based on the previous work by Davis, Putnam, and Robinson) in 1970 this question was answered negatively. This result has inspired a large amount of work on various generalizations of Hilbert's 10-th problem, where one searches for solutions in rings larger than Z, for instance over a number field. In general, a basic expectation is that the answer still remains negative, however for equations of small degree it is possible to suggest an algorithm. The approach that I want to discuss comes from arithmetic geometry and is based on the notion of height of points in a projective space. Starting from just a search for rational points on some very simple varieties, one can rather quickly get to some more general effective results related for instance to the algebraic theory of quadratic forms. In the first talk I will introduce the problem, the machinery, and discuss the case of linear equations. In the second talk, I will concentrate on the case of quadratic forms. If I time allows, I will also briefly discuss what can be done for higher degree.


This lecture was given in two parts during the Algebra/Number Theory/Combinatorics Seminar at the Claremont Colleges in September 2007. The first part of the talk was given on September 18th, while the second was given on September 25th.

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© 2007 Lenny Fukshansky

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