Sampling cubic rings

<p> We explain how to construct a uniformly random cubic integral domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of given signature with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue d i s c left-parenthesis upper S right-parenthesis EndAbsoluteValue less-than-or-equal-to upper T"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> | </mml:mo> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> <mml:mi>s</mml:mi> <mml:mi>c</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo fence="false" stretchy="false"> | </mml:mo> <mml:mo> ≤ </mml:mo> <mml:mi>T</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\lvert disc(S)\rvert \leq T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in expected time <inline-formula content-type="math/tex"> <tex-math>\widetilde \mathcal {O}(\log T)</tex-math> </inline-formula> . </p>