Bibliographic record
Abstract
Abstract For counting the zeros of a real polynomial f in an interval, one important approach was to study the behaviour of f(x) and that of certain auxiliary polynomials as x passes through a zero (see the theorems of Budan Fourier and Sturm and their proofs). The situation is quite different when we want to count the zeros of a complex polynomial f inside a simply connected domain Ω We do not know a path that passes through the zeros. However, since f has only a finite number of zeros, we may always replace Ω by an appropriate bounded domain Ω′ whose boundary is a Jordan curve, γ that surrounds the zeros to be counted. This suggests that we employ the principle of the argument (Theorem 1.1.3). It remains to study how the argument of f(z) changes as z traverses γ. For this purpose, we shall employ the concept of a Cauchy index. We start with a heuristic consideration.
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How this classification was reachedexpand
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) | 0.006 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".