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Polynomial products modulo primes and applications

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Abstract

For any polynomial \(P(x)\in {\mathbb {Z}}[x],\) we study arithmetic dynamical systems generated by \(\displaystyle {F_P(n)=\mathop {\prod \nolimits _{k\le n}}}P(k)(\text {mod}\ p),\)\(n\ge 1\). We apply this to improve the lower bound on the number of distinct quadratic fields of the form \({\mathbb {Q}}(\sqrt{F_P(n)})\) in short intervals \(M\le n\le M+H\) previously due to Cilleruelo, Luca, Quirós and Shparlinski. As a second application, we estimate the average number of missing values of \(F_P(n)(\text {mod}\ p)\) for special families of polynomials, generalizing previous work of Banks, Garaev, Luca, Schinzel, Shparlinski and others.

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Notes

  1. Improving this bound seems hard to the authors.

  2. These are a subset of the previous \(\alpha _i-j\).

  3. The irreducible case is in fact the worst. Indeed, if \(f_n\) has a lot of factors, it will produce many more roots modulo p and consequently many missing values.

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Acknowledgements

The authors would like to thank Andrew Granville, Andrzej Schinzel and Igor Shparlinski for valuable remarks. The first author is also grateful to Andrzej Schinzel for his hospitality during his visit to IMPAN (Warsaw).

Funding

The funding was provided bt Austrian Science Fund (Grant Number Y-901).

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Authors and Affiliations

  1. Max Planck Institute for Mathematics, Bonn, Germany

    Oleksiy Klurman

  2. 5010 Institut für Analysis und Zahlentheorie, Steyrergasse 30, 8010, Graz, Austria

    Marc Munsch

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  1. Oleksiy Klurman
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  2. Marc Munsch
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Correspondence to Marc Munsch.

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Communicated by Adrian Constantin.

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Marc Munsch greatly acknowledges support of the Austrian Science Fund (FWF), START-project Y-901 “Probabilistic methods in analysis and number theory” headed by Christoph Aistleitner.

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Klurman, O., Munsch, M. Polynomial products modulo primes and applications. Monatsh Math 191, 577–593 (2020). https://doi.org/10.1007/s00605-019-01359-6

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