Bounds on exponential sums and the polynomial waring problem mod p

Journal of the London Mathematical Society

Volumn 67, Issue 2, 2003, Pages 319-336

  Cochrane, Todd ^a   Pinner, Christopher ^a   Rosenhouse, Jason ^a  

^a KANSAS STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0037907638     PISSN: 00246107     EISSN: None     Source Type: Journal    
DOI: 10.1112/S0024610702004040     Document Type: Article

References (26)

1. N. M. AKULINICHEV, 'Estimates for rational trigonometric sums of a special type', Dokl. Acad. Sci. USSR 161 (1965) 743-745 (Russian), Doklady 161 (1965) 480-482 (English).
2. N. M. AKULINICHEV, 'Estimates for rational trigonometric sums of a special type', Dokl. Acad. Sci. USSR 161 (1965) 743-745 (Russian), Doklady 161 (1965) 480-482 (English).
3. J. D. BOVE, 'A new upper bound for Waring's problem (mod p)', Acta Arith. 32 (1977) 157-162.
4. L. CARLITZ, D. J. LEWIS, W. H. MILLS and E. G. STRAUS, 'Polynomials over finite fields with minimal value sets', Mathematika 8 (1961) 121-130.
5. I. CHOWLA, 'On Waring's problem (mod p)', Proc. Indian Nat. Acad. Part A 13 (1943) 195-200.
6. S. CHOWLA, H. B. MANN and E. G. STRAUS, 'Some applications of the Cauchy-Davenport theorem', Norske Vid. Selsk. Forh. Trondheim 32 (1959) 74-80.
7. E. DOBROWOLSKI, 'On a question of Lehmer and the number of irreducible factors of a polynomial', Acta Arith. 34 (1979) 391-401.
8. M. M. DODSON, 'On Waring's problem in GF[p]', Acta Arith. 19 (1971) 147-173.
9. M. M. DODSON and A. TIETÄVÄINEN, 'A note on Waring's problem in GF[p]', Acta Arith. 30 (1976) 159-167.
10. A. GARCIA and J. F. VOLOCH, 'Fermat curves over finite fields', J. Number Theory 30 (1988) 345-356.
11. G. H. HARDY and J. E. LITTLEWOOD, 'Some problems of "Partitio Numerorum". VIII: The number T(fc) in Waring's problem', Proc. London Math. Soc. (2) 28 (1927) 518-542.
12. D. R. HEATH-BROWN and S. KONYAGIN, 'New bounds for Gauss sums derived from kth powers, and for Heilbronn's exponential sum', Quart. J. Math. 51 (2000) 221-235.
13. 12. A. A. KARATSUBA, 'On estimates of complete trigonometric sums', Matem. Zametki 1 (1967) 199-208 (Russian), Math. Notes (1968) 133-139 (English).
14. B. S. KASHIN and A. A. SAAKYAN, Orthogonal series (Nauka, Moscow, 1984) (Russian), (American Mathematical Society, Providence, RI, 1989) (English).
15. S. V. KONYAGIN, 'On estimates of Gaussian sums and Waring's problem for a prime modulus', Trudy Mat. Inst. Steklov 198 (1992) 111-124 (Russian), Proc. Steklov Inst. Math. (1994) 105-107 (English).
16. S. V. KONYAGIN, 'Exponential sums over multiplicative groups of residues', preprint, Moscow State University, 2000.
17. S. V. KONYAGIN and I. E. SHPARLINSKI, Character sums with exponential functions and their applications (Cambridge University Press, Cambridge, 1999).
18. L. J. MORDELL, 'On a sum analogous to a Gauss's sum', Quant. J. Math. 3 (1932) 161-167.
19. G. L. MULLEN and I. E. SHPARLINSKI, 'Open problems and conjectures in finite fields', Finite fields and applications (Glasgow, 1995), London Mathematical Society Lecture Note Series 233 (ed. S. Cohen and H. Niederreiter, Cambridge University Press, Cambridge, 1996) 243-268.
20. C. POWELL, 'Bounds for multiplicative cosets over fields of prime order', Math. Comp. 218 (1997) 807-822.
21. I. E. SHPARLINSKI, 'On bounds of Gaussian sums', Matem. Zametki 50 (1991) 122-130 (Russian).
22. I. E. SHPARLINSKI, 'On Gaussian sums for finite fields and elliptic curves', Proceedings of the 1st French-Soviet Workshop on Algebraic Coding, Paris, 1991, Lecture Notes in Computer Science 537 (Springer, 1992) 5-15.
23. A. TIETÄVÄINEN, 'Proof of a conjecture of S. Chowla', J. Number Theory 1 (1975) 353-356.
24. J. D. VAALER, 'A geometric inequality with applications to linear forms', Pacific J. Math. 83 (1979) 543-553.
25. L. C. WASHINGTON, Introduction to cyclotomic fields, Graduate Texts in Mathematics 83 (Springer, New York, 1982).
26. A. WEIL, 'On some exponential sums', Proc. Nat. Acad. Sci. USA 34 (1948) 204-207.