Laurent Series and Puiseux Series in Maple
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Bibliographic record
Abstract
Let K be an algebraically closed field of characteristic zero. The field of fractions of the ring of formal multivariate power series over K, is called the field of formal multivariate Laurent series. In this document, we follow the ideas introduced by Monforte and Kauers in their paper Formal Laurent Series in Several Variables. Our objective is to report on a first implementation of formal multivariate Laurent series inside of Maple, and explain the challenges we had to overcome. In order to accomplish this goal, we make use of the already existing MultitivariatePowerSeries package, and its lazy evaluation scheme. In particular, we expose our ideas for adding and multiplying Laurent series with support inside different cones, where the support of a Laurent series is the set of all exponents of all non-zero monomials of our series. We also describe our biggest challenge, how to invert a Laurent series. Unfortunately, this problem cannot be completely solved in a lazy evaluation context. We describe some situations where we can solve the problem completely; our approach for the cases that fall outside of these situations; and how we let the user customize this approach, trading off between speed and the likelihood of an incorrect result.
 The algebraic closure of the field of formal multivariate Laurent series is call the field of formal multivariate Puiseux series. As an extension of our current work, we also present our ideas for an implementation of a multivariate Puiseux series object inside of Maple.
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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 | 0.001 |
| 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