Sampling cubic rings

TL;DR

This paper presents an efficient method to generate uniformly random cubic integral domains with bounded discriminant, achieving expected logarithmic time complexity.

Contribution

It introduces a novel algorithm for sampling cubic rings uniformly at random within a specified discriminant bound.

Findings

Algorithm runs in expected old ( ext{disc}(S)) ext{ time}

Achieves uniform sampling of cubic rings

Provides a practical method for probabilistic enumeration

Abstract

We explain how to construct a uniformly random cubic integral domain $S$ of given signature with $|\text{disc}(S)| \leq T$ in expected time $\widetilde O(\log T)$.