Unimodular completion of polynomial matrices
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Bibliographic record
Abstract
Given a rectangular matrix F ∈ K[x]mxn with m < n of univariate polynomials over a field K. we give an efficient algorithm for computing a unimodular completion of F. Our algorithm is deterministic and computes such a completion, when it exists, with cost O~ (nωs) field operations from K. Here s is the average of the m largest column degrees of F and ω is the exponent on the cost of matrix multiplication. Here O~ is big-O but with log factors removed. If a unimodular completion does not exist for F, our algorithm computes a unimodular completion for a right cofactor of a column basis of F, or equivalently, computes a completion that preserves the generalized determinant.
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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 | 0.000 | 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 |
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