Exact computation of Simultaneous Rational Approximants
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Bibliographic record
Abstract
In exact computing environments such as Maple and Mathematica problems often have symbolic parameters. As such a typical domain for computation is an integral domain (such as Q[a1,...,ak]) rather than a field. In such environments growth of coefficients in intermediate computations are a central concern. For methods that involve elimination intermediate growth can be controlled by removing greatest common divisors at each step. Fraction-free computation is an elimination process which controls coefficient growth in intermediate computations while at the same time avoids expensive greatest common divisor computations. In this talk we give a new, fast algorithm for solving the simultaneous Pad´ e approximation problem. The algorithm is fraction-free and is intended for computation in exact arithmetic. The algorithm gives significant improvement on previous fraction-free methods, in particular when solved via the use of vector Hermite-Pad´ e approximation using the FFFG order basis algorithm previously done by the authors. The improvements are both in terms of bit complexity and in reduced size of the intermediate quantities. The primary technique takes advantage of certain duality properties of Hermite-Pad´ e and SimultaneousPad´ e approximation problems.
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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