An Efficient Solution to Systems of Multivariate Polynomial Using Expression Trees
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Bibliographic record
Abstract
In recent years, several quite successful attempts have been made to solve systems of polynomial constraints, using geometric design tools, exploiting the availability of subdivision-based solvers [7], [11], [12], [15]. This broad range of methods includes both binary domain subdivision as well as the projected polyhedron method of Sherbrooke and Patrikalakis [15]. A prime obstacle in using subdivision solvers is their scalability. When the given constraint is represented as a tensor product of all its independent variables, it grows exponentially in size as a function of the number of variables. In this work, we show that for many applications, especially geometric ones, the exponential complexity of the constraints can be reduced to a polynomial by representing the underlying structure of the problem in the form of expression trees that represent the constraints. We demonstrate the applicability and scalability of this representation and compare its performance to that of tensor product constraint representation through several examples.
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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