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Applications of an elementary resolution of singularities algorithm to exponential sums and congruences modulo p n

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Abstract

Using the classical analysis resolution of singularities algorithm of [G4], we generalize the theorems of [G3] on R n sublevel set volumes and oscillatory integrals with real phase function to functions over an arbitrary local field of characteristic zero. The p-adic cases of our results provide new estimates for exponential sums as well as new bounds on how often a function f(x), such as a polynomial with integer coefficients, is divisible by various powers of a prime p when x is an integer. Unlike many papers on such exponential sums and p-adic oscillatory integrals, we do not require the Newton polyhedron of the phase to be nondegenerate, but rather as in [G3] we have conditions on the maximum order of the zeroes of certain polynomials corresponding to the compact faces of this Newton polyhedron.

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References

  1. V. Arnold, S Gusein-Zade and A. Varchenko, Singularities of Differentiable Maps. Volume II, Monographs in Mathematics, Vol. 83, Birkhäuser, Boston, MA, 1988.

    Book  Google Scholar 

  2. R. Cluckers, Analytic Van der Corput Lemma for p-adic oscillatory integrals, singular Fourier transforms, and restriction theorems, Expositiones Mathematicae 29 2011, 371–386.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Denef, Report on Igusa’s local zeta function. Séminaire Bourbaki, Vol. 1990/91, Astérisque No. 201–203 (1991), Exp. No. 741, 359–386 (1992).

    Google Scholar 

  4. J. Denef and K. Hoornaert, Newton polyhedra and Igusa’s local zeta function, Journal of Number Theory 89 2001, 31–64.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Denef and F. Loeser, Motivic Igusa zeta functions, Journal of Algebraic Geometry 7 1998, 505–537.

    MathSciNet  MATH  Google Scholar 

  6. F. Gouvea, p-adic Numbers. An Introduction, Second edition, Universitext, Springer-Verlag, Berlin, 1997.

    Book  MATH  Google Scholar 

  7. M. Greenblatt, A Coordinate-dependent local resolution of singularities and applications, Journal of Functional Analysis 255 2008, 1957–1994.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Greenblatt, Resolution of singularities, asymptotic expansions of oscillatory integrals, and related phenomena, Journal d’Analyse Mathématique 111 2011, 221–245.

    MathSciNet  MATH  Google Scholar 

  9. M. Greenblatt, Oscillatory integral decay, sublevel set growth, and the Newton polyhedron, Mathematische Annalen 346 2010, 857–895.

    MathSciNet  MATH  Google Scholar 

  10. M. Greenblatt, A constructive elementary method for local resolution of singularities, submitted.

  11. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, Annals of Mathematics 79 1964, 109–203.

    Article  MathSciNet  MATH  Google Scholar 

  12. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. II, Annals of Mathematics 79 1964, 205–326.

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Igusa, Complex powers and asymptotic expansions. I, Journal für die Reine und Agnewandte Mathematik 268/269 1974, 110–130.

    MathSciNet  MATH  Google Scholar 

  14. J. Igusa, Complex powers and asymptotic expansions. II, Journal für die Reine und Agnewandte Mathematik 278/279 1975, 307–321.

    MathSciNet  MATH  Google Scholar 

  15. J. Igusa, Forms of Higher Degree, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Vol. 59, Tata Institute of Fundamental Research, Bombay; Narosa, New Delhi, 1978.

    Google Scholar 

  16. B. Lichtin and D. Meuser, Poles of local zeta functions and Newton polygons, Compositio Mathematica 55 1985, 313–332.

    MathSciNet  MATH  Google Scholar 

  17. K. Rogers, A van der Corput lemma for the p-adic numbers, Proceedings of the American Mathematical Society 133 2005, 3525–3534.

    Article  MathSciNet  MATH  Google Scholar 

  18. A. N. Varchenko, Newton polyhedra and estimation of oscillating integrals, Functional Analysis and its Applications 18 1976, 175–196.

    MATH  Google Scholar 

  19. W. Veys, Determination of the poles of the topological zeta function for curves, Manuscripta Mathematica 87 1995, 435–448.

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Wright, Exponential sums and polynomial congruences in two variables: the quasihomogeneous case, preprint.

  21. W. Zuniga-Galindo, Igusa’s local zeta functions of semiquasihomogeneous polynomials, Transactions of the American Mathematical Society 353 2001, 3193–3207.

    Article  MathSciNet  MATH  Google Scholar 

  22. W. Zuniga-Galindo, Local zeta functions and Newton polyhedra, Nagoya Mathematical Journal 172 2003, 31–58.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 Science and Engineering Offices, 851 S. Morgan Street, Chicago, IL, 60607-7045, USA

    Michael Greenblatt

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  1. Michael Greenblatt
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Correspondence to Michael Greenblatt.

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This research was supported in part by NSF grants DMS-0919713 and DMS-1001070.

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Greenblatt, M. Applications of an elementary resolution of singularities algorithm to exponential sums and congruences modulo p n . Isr. J. Math. 212, 315–335 (2016). https://doi.org/10.1007/s11856-016-1288-7

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  • DOI: https://doi.org/10.1007/s11856-016-1288-7

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