Drinfeld Modules with Complex Multiplication, Hasse Invariants and\n Factoring Polynomials over Finite Fields
Bibliographic record
Abstract
We present a novel randomized algorithm to factor polynomials over a finite\nfield $\\F_q$ of odd characteristic using rank $2$ Drinfeld modules with complex\nmultiplication. The main idea is to compute a lift of the Hasse invariant\n(modulo the polynomial $f \\in \\F_q[x]$ to be factored) with respect to a random\nDrinfeld module $\\phi$ with complex multiplication. Factors of $f$ supported on\nprime ideals with supersingular reduction at $\\phi$ have vanishing Hasse\ninvariant and can be separated from the rest. Incorporating a Drinfeld module\nanalogue of Deligne's congruence, we devise an algorithm to compute the Hasse\ninvariant lift, which turns out to be the crux of our algorithm. The resulting\nexpected runtime of $n^{3/2+\\varepsilon} (\\log q)^{1+o(1)}+n^{1+\\varepsilon}\n(\\log q)^{2+o(1)}$ to factor polynomials of degree $n$ over $\\F_q$ matches the\nfastest previously known algorithm, the Kedlaya-Umans implementation of the\nKaltofen-Shoup algorithm.\n
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How this classification was reachedexpand
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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".