Polynomial parametrization of quadratic Diophantine equations: decision and construction
We give an algorithm that, for any polynomial with integer coefficients of degree at most two, decides whether its integer zero set is a finite union of polynomial families with integer coefficients and constructs such families whenever they exist. The main step is a constructive proof that, for every nondegenerate integral quadratic form \(Q\) isotropic over \(\mathbb{Q}\) and every integer \(m\), the integer solutions of \(Q(\mathbf{x})=m\) form a finite union of polynomial families. This proves the nondegenerate case of a result announced without proof by Vaserstein. For fixed \(Q\), the…
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