Linear Hensel Lifting for Zp[x,y] for n Factors with Cubic Cost
Bibliographic record
Abstract
We present a new algorithm for performing linear Hensel lifting on bivariate polynomials over the finite field Zp for some prime p. Our algorithm lifts n monic, univariate polynomials to recover the factors of a polynomial A(x,y) in Zp[x,y] which is monic in x, and bounded by degrees dx = deg(A,x) and dy = deg(A,y). Our algorithm improves upon Bernardin's algorithm in [1] and reduces the number of arithmetic operations in Zp from O(n dx^2 dy^2) to O(dx^2 dy + dx dy^2) for p >= dx. Experimental results in C verify that our algorithm compares favorably with Bernardin's for large degree polynomials. Moreover, we've implemented a Quadratic Hensel lifting algorithm in Magma to show that our cubic Linear Hensel lifting algorithm outperforms Magma's Quadratic Hensel lifting for a wide range of input sizes.
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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 | 0.000 |
| 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".