Uniform estimates for almost primes over finite fields
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- by Dor Elboim and Ofir Gorodetsky PDF
- Proc. Amer. Math. Soc. 150 (2022), 2807-2822 Request permission
Abstract:
We establish a new asymptotic formula for the number of polynomials of degree $n$ with $k$ prime factors over a finite field $\mathbb {F}_q$. The error term tends to $0$ uniformly in $n$ and in $q$. Previously, asymptotic formulas were known either for fixed $q$, through the works of Warlimont [Arch. Math. (Basel) 60 (1993), pp. 58–72] and Hwang [Random Structures Algorithms 13 (1998), pp. 17–47], or for small $k$, through the work of Arratia, Barbour and Tavaré [Math. Proc. Cambridge Philos. Soc. 114 (1993), pp. 347–368].
As an application, we estimate the total variation distance between the number of cycles in a random permutation on $n$ elements and the number of prime factors of a random polynomial of degree $n$ over $\mathbb {F}_q$. The distance tends to $0$ at rate $1/(q\sqrt {\log n})$. Previously this was only understood when either $q$ is fixed and $n$ tends to $\infty$, or $n$ is fixed and $q$ tends to $\infty$, by results of Arratia, Barbour and Tavaré.
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Additional Information
- Dor Elboim
- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
- MR Author ID: 1326512
- Email: delboim@math.princeton.edu
- Ofir Gorodetsky
- Affiliation: Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom
- MR Author ID: 1234845
- ORCID: 0000-0002-1435-9650
- Email: ofir.goro@gmail.com
- Received by editor(s): August 24, 2020
- Received by editor(s) in revised form: October 12, 2021
- Published electronically: March 29, 2022
- Additional Notes: The second author was supported by the European Research Council (ERC) under the European Union’s 2020 research and innovation programme (ERC grant agreement nos 786758 and 851318)
- Communicated by: Matthew A. Papanikolas
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 2807-2822
- MSC (2020): Primary 11K65; Secondary 11T06, 05A05
- DOI: https://doi.org/10.1090/proc/15870
- MathSciNet review: 4428869