CSUN Algebra, Number Theory, and Discrete Mathematics Seminar

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.