Igusa’s Local Zeta Functions and Exponential Sums for Arithmetically Non Degenerate Polynomials
Journal de Théorie des Nombres de Bordeaux, Tome 30 (2018) no. 1, pp. 331-354.

Nous étudions la fonction zêta locale tordue associée à un polynôme en deux indéterminées à coefficients dans un corps local non archimédien de caractéristique arbitraire. Sous l’hypothèse que le polynôme est arithmétiquement non-dégénéré, nous obtenous une liste explicite de candidats pour les pôles en termes des données géométriques obtenues à partir d’une famille de polygones de Newton arithmétiques attachés au polynôme. La notion de non-dégénérescence arithmétique de Saia et Zúñiga-Galindo est plus faible que la notion habituelle de non-dégénérescence de Kouchnirenko. Finalement, on applique nos résultats pour obtenir des développements asymptotiques pour certaines sommes exponentielles associées à ces polynômes.

We study the twisted local zeta function associated to a polynomial in two variables with coefficients in a non-Archimedean local field of arbitrary characteristic. Under the hypothesis that the polynomial is arithmetically non degenerate, we obtain an explicit list of candidates for the poles in terms of geometric data obtained from a family of arithmetic Newton polygons attached to the polynomial. The notion of arithmetical non degeneracy due to Saia and Zúñiga-Galindo is weaker than the usual notion of non degeneracy due to Kouchnirenko. As an application we obtain asymptotic expansions for certain exponential sums attached to these polynomials.

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DOI : https://doi.org/10.5802/jtnb.1028
Classification : 11S40,  14G10,  11T23,  14M25
Mots clés : Igusa’s zeta functions, degenerate curves, Newton polygons, non-degeneracy conditions, exponential sums mod p m
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Adriana A. Albarracín-Mantilla; Edwin León-Cardenal. Igusa’s Local Zeta Functions and Exponential Sums for Arithmetically Non Degenerate Polynomials. Journal de Théorie des Nombres de Bordeaux, Tome 30 (2018) no. 1, pp. 331-354. doi : 10.5802/jtnb.1028. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1028/

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