Asymptotic distributions of the number of zeros of random polynomials in Hayes equivalence class over a finite field
Bibliographic record
Abstract
Hayes equivalence is defined on monic polynomials over a finite field F q in terms of the prescribed leading coefficients and the residue classes modulo a given monic polynomial Q . We study the distribution of the number of zeros in a random polynomial over finite fields in a given Hayes equivalence class. It is well known that the number of distinct zeros of a random polynomial over F q is asymptotically Poisson with mean 1. We show that this is also true for random polynomials in any given Hayes equivalence class. Asymptotic formulas are also given for the number of such polynomials when the degree of such polynomials is proportional to q and the degree of Q and the number of prescribed leading coefficients are bounded by q . When Q = 1 , the problem is equivalent to the study of the distance distribution in Reed-Solomon codes. Our asymptotic formulas extend some earlier results and imply that all words for a large family of Reed-Solomon codes are ordinary, which further supports the well-known Deep-Hole Conjecture.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.025 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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 source (direct Gemma or distilled Codex), 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".