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Large sieve estimate for multivariate polynomial moduli and applications

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Abstract

We prove large sieve inequalities with multivariate polynomial moduli and deduce a general Bombieri–Vinogradov type theorem for a class of polynomial moduli having a sufficient number of variables compared to its degree. This sharpens previous results of the first author in two aspects: the range of the moduli as well as the class of polynomials which can be handled. As a consequence, we deduce that there exist infinitely many primes p such that \(p-1\) has a prime divisor of size \(\gg p^{2/5+o(1)}\) that is the value of an incomplete norm form polynomial.

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Notes

  1. We could have stated our results removing repetitions or weighting every moduli by the inverse of the number of appearances. For our purpose these formulations are essentially equivalent due to the polynomials being considered in applications where we essentially have very small preimages.

  2. The result is slightly more complicated to state and gives the expected number of solutions to multidimensional Vinogradov systems. The correcting factor for applications depends also on the number of variables \(\ell \) but is very close to 2.

  3. Again the correcting factor is more complicated and depends on the number of variables but is just slightly larger than 2. For a more precise description of the gain here, see [23, Theorem 3.2] and the enlightening discussion following equation (1.4) in the same paper. For our purpose and to simplify the exposition we correct by a factor 2.

  4. See also the discussion on mathoverflow: https://mathoverflow.net/questions/68437/the-divisor-bound-in-number-fields.

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Acknowledgements

The second author would like to thank the Max Planck Institute for Mathematics, Bonn, and the University of Düsseldorf for support and hospitality during his work on this project. The second author also acknowledges support of the Austrian Science Fund (FWF), stand-alone project P 33043 “Character sums, L-functions and applications”.

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Authors and Affiliations

  1. Mathematisches Institut der Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225, Düsseldorf, Germany

    Karin Halupczok

  2. Universitá degli Studi di Genova, Dipartimento di Matematica, Via Dodecaneso 35, 16146, Genova, Italia

    Marc Munsch

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  1. Karin Halupczok
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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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Halupczok, K., Munsch, M. Large sieve estimate for multivariate polynomial moduli and applications. Monatsh Math 197, 463–478 (2022). https://doi.org/10.1007/s00605-021-01641-6

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