A đ-adic algorithm to compute the Hilbert class polynomial
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Abstract
Classically, the Hilbert class polynomial <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript normal upper Delta Baseline element-of bold upper Z left-bracket upper X right-bracket"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> Î </mml:mi> </mml:mrow> </mml:msub> <mml:mo> â </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">P_{\Delta }\in \mathbf {Z} [X]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of an imaginary quadratic discriminant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta"> <mml:semantics> <mml:mi mathvariant="normal"> Î </mml:mi> <mml:annotation encoding="application/x-tex">\Delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is computed using complex analytic techniques. In 2002, Couveignes and Henocq suggested a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic algorithm to compute <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript normal upper Delta"> <mml:semantics> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> Î </mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">P_{\Delta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Unlike the complex analytic method, it does not suffer from problems caused by rounding errors. In this paper we give a detailed description of the algorithm in the paper by Couveignes and Henocq, and our careful study of the complexity shows that, if the Generalized Riemann Hypothesis holds true, the expected runtime of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic algorithm is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis StartAbsoluteValue normal upper Delta EndAbsoluteValue left-parenthesis log StartAbsoluteValue normal upper Delta EndAbsoluteValue right-parenthesis Superscript 8 plus epsilon Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi mathvariant="normal"> Î </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>log</mml:mi> <mml:mo> ⥠</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi mathvariant="normal"> Î </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>8</mml:mn> <mml:mo>+</mml:mo> <mml:mi> Δ </mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(|\Delta |(\log |\Delta |)^{8+\varepsilon })</mml:annotation> </mml:semantics> </mml:math> </inline-formula> instead of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis StartAbsoluteValue normal upper Delta EndAbsoluteValue Superscript 1 plus epsilon Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi mathvariant="normal"> Î </mml:mi> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi> Δ </mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(|\Delta |^{1+\varepsilon })</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We illustrate the algorithm by computing the polynomial <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript negative 639"> <mml:semantics> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> â </mml:mo> <mml:mn>639</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">P_{-639}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> using a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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