Generalized Gorshkov–Wirsing Polynomials and the Integer Chebyshev Problem
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Bibliographic record
Abstract
Abstract. The Integer Chebyshev Problem is the problem of finding an inte-ger polynomial of degree n such that the supremum norm on [0, 1] is minimized. The most common technique used to find upper bounds is by explicit construc-tion of an example. This is often (although not always) done by heavy com-putational use of the LLL and the Simplex method. Among the first methods developed to find lower bounds was through a sequence of polynomials known as the Gorshkov-Wirsing polynomials. This paper studies properties of the Gorshkov-Wirsing polynomials. It is shown how to construct generalized Gorshkov-Wirsing polynomials on any interval [a, b], with a, b ∈ Q. An extensive search for generalized Gorshkov-Wirsing polynomials is done for a large family of [a, b]. Using generalized Gorshkov-Wirsing polynomials, LLL and the Simplex method, upper and lower bounds for the Integer Chebyshev Constant on intervals other than [0,1] are calculated. These methods are compared with other existing methods. 1.
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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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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