Tabulation of cubic function fields via polynomial binary cubic forms
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Bibliographic record
Abstract
We present a method for tabulating all cubic function fields over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F Subscript q Baseline left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {F}_q(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose discriminant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has either odd degree or even degree and the leading coefficient of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus 3 upper D"> <mml:semantics> <mml:mrow> <mml:mo> − </mml:mo> <mml:mn>3</mml:mn> <mml:mi>D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">-3D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a non-square in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F Subscript q Superscript asterisk"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>q</mml:mi> </mml:mrow> <mml:mo> ∗ </mml:mo> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathbb {F}_{q}^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , up to a given bound <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="degree left-parenthesis upper D right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>deg</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\deg (D)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Our method is based on a generalization of Belabas’ method for tabulating cubic number fields. The main theoretical ingredient is a generalization of a theorem of Davenport and Heilbronn to cubic function fields, along with a reduction theory for binary cubic forms that provides an efficient way to compute equivalence classes of binary cubic forms. The algorithm requires <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis upper B Superscript 4 Baseline q Superscript upper B Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>B</mml:mi> <mml:mn>4</mml:mn> </mml:msup> <mml:msup> <mml:mi>q</mml:mi> <mml:mi>B</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(B^4 q^B)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> field operations as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">B \rightarrow \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The algorithm, examples and numerical data for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q equals 5 comma 7 comma 11 comma 13"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> <mml:mo>,</mml:mo> <mml:mn>7</mml:mn> <mml:mo>,</mml:mo> <mml:mn>11</mml:mn> <mml:mo>,</mml:mo> <mml:mn>13</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">q=5,7,11,13</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are included.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it