Bit complexity for multi-homogeneous polynomial system solving\n Application to polynomial minimization
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Bibliographic record
Abstract
Multi-homogeneous polynomial systems arise in many applications. We provide\nbit complexity estimates for solving them which, up to a few extra other\nfactors, are quadratic in the number of solutions and linear in the height of\nthe input system under some genericity assumptions. The assumptions essentially\nimply that the Jacobian matrix of the system under study has maximal rank at\nthe solution set and that this solution set if finite. The algorithm is\nprobabilistic and a probability analysis is provided. Next, we apply these\nresults to the problem of optimizing a linear map on the real trace of an\nalgebraic set. Under some genericity assumptions, we provide bit complexity\nestimates for solving this polynomial minimization problem.\n
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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 | 0.000 |
| 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 it