Solving quadratic equations over Q-bar

09/08/2015 - 12:15pm
09/08/2015 - 1:10pm
Lenny Fukshansky (CMC)

Given a quadratic equation in N variables over a fixed number field K, there exists an algorithm to determine whether it has a non-trivial solution over K and to find such a solution. This matter becomes considerably more complicated for a system of quadratic equations: existence of a general such algorithm would contradict Matijasevich's negative answer to Hilbert's 10th problem. The problem however is more tractable if we allow searching in extensions of K. I will discuss an approach to this problem, which involves height functions, the common tools of Diophantine geometry. Our investigation extends previous results on small-height zeros of quadratic forms, including Cassels' theorem and its various generalizations and contributes to the literature of so-called "absolute" Diophantine results with respect to height.

Millikan 2099, Pomona College
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Organizational meeting in the same room preceding the talk at 12:00 noon.