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An improved Mordell type bound for exponential sums
Author(s):
Todd
Cochrane;
Christopher
Pinner
Journal:
Proc. Amer. Math. Soc.
133
(2005),
313-320.
MSC (2000):
Primary 11L07, 11L03
Posted:
September 2, 2004
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Abstract:
For a sparse polynomial ![$f(x)=\sum_{i=1}^r a_ix^{k_i}\in \mathbb Z [x]$](/proc/2005-133-02/S0002-9939-04-07726-3/gif-abstract0/img1.gif) , with  and  , we show that
thus improving upon a bound of Mordell. Analogous results are obtained for Laurent polynomials and for mixed exponential sums.
References:
-
- 1.
- N. M. Akulinichev, Estimates for rational trigonometric sums of a special type, Doklady Acad. Sci. USSR 161 (1965), 743-745. English translation in Doklady 161, no. 4 (1965), 480-482. MR 0177956 (31:2214)
- 2.
- F. N. Castro & C. J. Moreno, Mixed exponential sums over finite fields, Proc. Amer. Math. Soc. 128, no. 9 (2000), 2529-2537. MR 1690978 (2000m:11070)
- 3.
- T. Cochrane & C. Pinner, Stepanov's method applied to binomial exponential sums, Quart. J. Math. 54, no. 3 (2003), 243-255. MR 2013138
- 4.
- T. Cochrane, C. Pinner & J. Rosenhouse, Bounds on exponential sums and the polynomial Waring's problem mod p, J. London Math. Soc. (2) 67, no. 2 (2003), 319-336. MR 1956138 (2003m:11129)
- 5.
- H. Davenport, On certain exponential sums, Journal für Math., 169 (1933), 158-176.
- 6.
- D. R. Heath-Brown & S. Konyagin, New bounds for Gauss sums derived from
 th powers, and for Heilbronn's exponential sum, Quart. J. Math. 51 (2000), 221-235. MR 1765792 (2001h:11106) - 7.
- A. A. Karatsuba, On estimates of complete trigonometric sums, Matem. Zametki. 1 (1967), 199-208 (Russian); translation in Math. Notes (1968), 133-139.
- 8.
- S. Konyagin & I. Shparlinski, Character Sums with Exponential Functions and their Applications, Cambridge Univ. Press, Cambridge, 1999. MR 1725241 (2000h:11089)
- 9.
- L. J. Mordell, On a sum analogous to a Gauss's sum, Quart. J. Math. 3 (1932), 161-167.
- 10.
- G. I. Perel'muter, Estimate of a sum along an algebraic curve, Mat. Zametki 5 (1969), 373-380. MR 0241424 (39:2764)
- 11.
- I. E. Shparlinski, Computational and Algorithmic Problems in Finite Fields, Mathematics and its applications (Soviet series), Kluwer, 1992. MR 1249064 (94j:11122)
- 12.
- A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204-207. MR 0027006 (10:234e)
- 13.
- T. Wooley, A note on simultaneous congruences, J. Number Theory 58 (1996), no. 2, 288-297. MR 1393617 (97h:11037)
- 14.
- H. B. Yu, Estimates for complete exponential sums of special types, Math. Proc. Camb. Phil. Soc. 131 (2001), 321-326. MR 1857123 (2002g:11126)
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Additional Information:
Todd
Cochrane
Affiliation:
Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
Email:
cochrane@math.ksu.edu
Christopher
Pinner
Affiliation:
Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
Email:
pinner@math.ksu.edu
DOI:
10.1090/S0002-9939-04-07726-3
PII:
S 0002-9939(04)07726-3
Keywords:
Exponential sums
Received by editor(s):
July 23, 2002
Received by editor(s) in revised form:
September 6, 2002
Posted:
September 2, 2004
Communicated by:
Wen-Ching Winnie Li
Copyright of article:
Copyright
2004,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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