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Le poids des polynômes irréductibles à coefficients dans un corps fini

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Journal d'Analyse Mathématique Aims and scope

Résumé

Ce travail concerne le poids des polynômes irréductibles sur un corps fini, c’est à dire le nombre de coefficients non nuls de ces polynômes. Nous introduisons un analogue polynomial de la méthode de Vinogradov développée par Gallagher et Vaughan afin de majorer les sommes d’exponentielles associées. Cela nous permet d’une part d’étudier la répartition dans les progressions arithmétiques du poids des polynômes irréductibles et d’autre part de donner une estimation asymptotique (avec terme d’erreur) du nombre de polynômes irréductibles de degré fixé ayant un poids donné proche de la valeur moyenne.

Abstract

This work concerns the weight of irreducible polynomials over a finite field, i.e., the number of non-zero coefficients of these polynomials. We introduce a polynomial analog of the Vinogradov’s method developed by Gallagher and Vaughan, which leads to upper bounds for associated exponential sums. This allows us to study the distribution of the weight of irreducible polynomials in arithmetic progressions and to provide an asymptotic estimate (with an error term) for the number of irreducible polynomials of given degree whose weight is close to the expected value.

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Authors and Affiliations

  1. Institut de Mathématiques de Marseille, CNRS, UMR 7373, Aix-Marseille Université, CMI, 39, rue F. Joliot-Curie, 13453, Marseille Cedex 13, France

    Mireille Car

  2. Institut de Mathématiques de Marseille CNRS, UMR 7373, Aix-Marseille Université, 163 avenue de Luminy, Case 907, F-13288, Marseille Cedex 9, France

    Christian Mauduit

Authors
  1. Mireille Car
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  2. Christian Mauduit
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Correspondence to Mireille Car.

Additional information

Christian Mauduit has sadly passed away before the publication of this paper.

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Car, M., Mauduit, C. Le poids des polynômes irréductibles à coefficients dans un corps fini. JAMA 146, 441–486 (2022). https://doi.org/10.1007/s11854-022-0199-2

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  • DOI: https://doi.org/10.1007/s11854-022-0199-2

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