{"id":"hexagon/2610.00652","pdf":"https://hexagonmath.org/pdf/2610.00652v1","url":"https://hexagonmath.org/abs/2610.00652v1","tags":["msc:11D09","msc:11E12","msc:11E57","msc:11R11","msc:20G30","arithmetic groups","integral points","polynomial parametrization","quadratic fields","quadratic forms"],"score":0,"title":"Polynomial parametrization of quadratic Diophantine equations: decision and construction","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Bogdan Grechuk","Ashleigh Ratcliffe"],"reviews":0,"subject":"math.NT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT","Claude Opus"],"discussed":0,"published":"2026-10-29","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}