The nearest polynomial with a given zero, revisited
Why this work is in the frame
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Bibliographic record
Abstract
In his 1999 SIGSAM BULLETIN paper [7], H. J. Stetter gave an explicit formula for finding the nearest polynomial with a given zero. This present paper revisits the issue, correcting a minor omission from Stetter's formula and explicitly extending the results to different polynomial bases.Experiments with our implementation demonstrate that the formula may not after all, fully solve the problem, and we discuss some outstanding issues: first, that the nearest polynomial with the given zero may be identically zero (which might be surprising), and, second, that the problem of finding the nearest polynomial of the same degree with a given zero may not, in fact, have a solution. A third variant of the problem, namely to find the nearest monic polynomial (given a monic polynomial initially) with a given zero, a problem that makes sense in some polynomial bases but not others, can also be solved with Stetter's formula, and this may be more satisfactory in some circumstances. This last can be generalized to the case where some coefficients are intrinsic and not to be changed, whereas others are empiric and may safely be changed. Of course, this minor generalization is implicit in [7]; This paper simply makes it explicit.
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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 | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.001 |
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