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A NOTE ON SIMULTANEOUS CONGRUENCES, II: MORDELL REVISED

Part of: Diophantine equations

Published online by Cambridge University Press:  25 March 2010

TREVOR D. WOOLEY
TREVOR D. WOOLEY*
Affiliation:
School of Mathematics, University of Bristol, University Walk, Clifton, Bristol BS8 1TW, UK (email: [email protected], [email protected])
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Abstract

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When p is a prime number, and k1,…,kt are natural numbers with 1≤k1<k2<⋯<kt<p, we show that the simultaneous congruences ∑ t1xkji≡∑ t1ykjimod p (1≤j≤t) possess at most k1⋯ktpt solutions with 1≤xi,yi≤p (1≤i≤t). Analogous conclusions are provided when one or more of the exponents ki is negative.

Keywords

congruences in many variablesexponential sums

MSC classification

Secondary: 11D79: Congruences in many variables
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 88 , Issue 2 , April 2010 , pp. 261 - 275
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Akulinchev, N. M., ‘Bounds for rational trigonometric sums of a special type’, Dokl. Akad. Nauk SSSR 161 (1965), 743–745.Google Scholar
[2]Bourgain, J., ‘Mordell’s exponential sum estimate revisited’, J. Amer. Math. Soc. 18 (2005), 477–499.CrossRefGoogle Scholar
[3]Cafure, A. and Matera, G., ‘Improved explicit estimates on the number of solutions of equations over a finite field’, Finite Fields Appl. 12 (2006), 155–185.CrossRefGoogle Scholar
[4]Cochrane, T., Coffelt, J. and Pinner, C., ‘A further refinement of Mordell’s bound on exponential sums’, Acta Arith. 116 (2005), 35–41.CrossRefGoogle Scholar
[5]Cochrane, T., Coffelt, J. and Pinner, C., ‘A system of simultaneous congruences arising from trinomial exponential sums’, J. Théor. Nombres Bordeaux 18 (2006), 59–72.CrossRefGoogle Scholar
[6]Cochrane, T. and Pinner, C., ‘An improved Mordell type bound for exponential sums’, Proc. Amer. Math. Soc. 133 (2005), 313–320.Google Scholar
[7]Cochrane, T., Pinner, C. and Rosenhouse, J., ‘Bounds on exponential sums and the polynomial Waring problem mod p’, J. Lond. Math. Soc. (2) 67 (2003), 319–336.CrossRefGoogle Scholar
[8]Hartshorne, R., Algebraic Geometry (Springer, Berlin, 1977).Google Scholar
[9]Heintz, J. and Schnorr, C.-P., ‘Testing polynomials which are easy to compute’, in: Logic and Algorithmic (Zurich, 1980), Monographies de l’Enseignement Mathématique, 30 (Univ. Genève, Geneva, 1982), pp. 237–254.Google Scholar
[10]Karatsuba, A. A., ‘Estimates of complete trigonometric sums’, Mat. Zametki 1 (1967), 199–208.Google Scholar
[11]Mordell, L. J., ‘On a sum analogous to Gauss’s sum’, Q. J. Math. Oxford 3 (1932), 161–167.CrossRefGoogle Scholar
[12]Vinogradov, I. M., ‘New estimates for Weyl sums’, Dokl. Akad. Nauk SSSR 8 (1935), 195–198.Google Scholar
[13]Weil, A., ‘On some exponential sums’, Proc. Natl. Acad. Sci. USA 34 (1948), 204–207.CrossRefGoogle ScholarPubMed
[14]Wooley, T. D., ‘On simultaneous additive equations, III’, Mathematika 37 (1990), 85–96.Google Scholar
[15]Wooley, T. D., ‘A note on simultaneous congruences’, J. Number Theory 58 (1996), 288–297.CrossRefGoogle Scholar
[16]Yu, H. B., ‘Estimates for complete exponential sums of special types’, Math. Proc. Cambridge Philos. Soc. 131 (2001), 321–326.Google Scholar