A polynomial analogue of Jacobsthal function
- 作者: Kalmynin A.1,2, Konyagin S.1
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隶属关系:
- Steklov Mathematical Institute of Russian Academy of Sciences
- National Research University Higher School of Economics
- 期: 卷 88, 编号 2 (2024)
- 页面: 33-43
- 栏目: Articles
- URL: https://journals.rcsi.science/1607-0046/article/view/254262
- DOI: https://doi.org/10.4213/im9467
- ID: 254262
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详细
For a polynomial $f(x)\in \mathbb Z[x]$ we study an analogue of Jacobsthal function defined by$j_f(N) =\max_{m}\{for some x \in \mathbb N$ the inequality$(x+f(i),N) >1 $ holds for all $i \leqslant m\}$.We prove the lower bound$$j_f(P(y))\gg y(\ln y)^{\ell_f-1}(\frac{(\ln\ln y)^2}{\ln\ln\ln y})^{h_f}(\frac{\ln y\ln\ln\ln y}{(\ln\ln y)^2})^{M(f)},$$where $P(y)$ is the product of all primes $p$ below $y$, $\ell_f$ is the number of distinct linear factors of $f(x)$, $h_f$ is the number of distinct non-linear irreducible factors and $M(f)$ is the average size of the maximal preimage of a point under a map $f\colon \mathbb F_p\to \mathbb F_p$. The quantity $M(f)$ is computed in terms of certain Galois groups.
作者简介
Aleksandr Kalmynin
Steklov Mathematical Institute of Russian Academy of Sciences; National Research University Higher School of Economics
Email: alkalb1995cd@mail.ru
Scopus 作者 ID: 57189372991
Researcher ID: AAG-4815-2019
without scientific degree
Sergei Konyagin
Steklov Mathematical Institute of Russian Academy of Sciences
Email: konyagin23@gmail.com
ORCID iD: 0000-0002-9669-5446
Scopus 作者 ID: 6701482885
Researcher ID: Q-4807-2016
Doctor of physico-mathematical sciences, Professor
参考
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