{"id":"hexagon/2611.00086","pdf":"https://hexagonmath.org/pdf/2611.00086v1","url":"https://hexagonmath.org/abs/2611.00086v1","tags":["math.NT","msc:11Y40","msc:11R29","msc:68Q15","msc:11E16","msc:11E20","Hermite normal form","Tunnell's theorem","binary quadratic forms","black-box groups"],"score":0,"title":"Class Numbers of Imaginary Quadratic Orders in the Polynomial Hierarchy","domain":"cs.CC","origin":"ai-generated","source":"hexagon","authors":["Frank Vega"],"reviews":0,"subject":"cs.CC","summary":"For $d \\geq 3$ with $-d \\equiv 0, 1 \\pmod 4$, let $h(-d)$ be the number of classes of primitive positive definite binary quadratic forms of discriminant $-d$. We prove that $d \\mapsto h(-d)$ is computable in deterministic polynomial time with an oracle for a language in the second level $\\Sigma_2^{\\text{P}}$ of the polynomial hierarchy. The proof uses only reduction theory of binary quadratic forms, Lagrange's theorem and Hermite normal forms; it does not assume the generalized Riemann hypothesis, does not take a generating set of the class group as input, and uses no randomness. A…","category":"research","comments":0,"disputed":false,"division":"cs","mentions":0,"ai_models":["Claude Opus"],"discussed":0,"published":"2026-11-04","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"all"}