Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-05-19T12:12:17.924Z Has data issue: false hasContentIssue false
  • English
  • Français

HUA-TYPE ITERATION FOR MULTIDIMENSIONAL WEYL SUMS

Part of: Additive number theory; partitions Diophantine equations Exponential sums and character sums

Published online by Cambridge University Press:  27 March 2012

Scott T. Parsell
Scott T. Parsell*
Affiliation:
Department of Mathematics, West Chester University, 25 University Ave., West Chester, PA 19383, U.S.A. (email: sparsell@wcupa.edu)
Get access

Abstract

We develop Weyl differencing and Hua-type lemmata for a class of multidimensional exponential sums. We then apply our estimates to bound the number of variables required to establish an asymptotic formula for the number of solutions of a system of diophantine equations arising from the study of linear spaces on hypersurfaces. For small values of the degree and dimension, our results are superior to those stemming from the author’s earlier work on Vinogradov’s mean value theorem.

MSC classification

Secondary: 11D45: Counting solutions of Diophantine equations 11D72: Equations in many variables 11L07: Estimates on exponential sums 11L15: Weyl sums 11P55: Applications of the Hardy-Littlewood method
Type
Research Article
Information
Mathematika , Volume 58 , Issue 2 , July 2012 , pp. 209 - 224
Copyright
Copyright © University College London 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Arkhipov, G. I., Karatsuba, A. A. and Chubarikov, V. N., Multiple trigonometric sums. Tr. Mat. Inst. Steklov 151 (1980), 1–126.Google Scholar
[2]Baker, R. C., Diophantine Inequalities, Clarendon Press (Oxford, 1986).Google Scholar
[3]Birch, B. J., Homogeneous forms of odd degree in a large number of variables. Mathematika 4 (1957), 102–105.CrossRefGoogle Scholar
[4]Birch, B. J., Forms in many variables. Proc. R. Soc. Lond. Ser. A 265 (1961), 245–263.Google Scholar
[5]Boklan, K., The asymptotic formula in Waring’s problem. Mathematika 41 (1994), 329–347.CrossRefGoogle Scholar
[6]Brauer, R., A note on systems of homogeneous algebraic equations. Bull. Amer. Math. Soc. 51 (1945), 749–755.CrossRefGoogle Scholar
[7]Brüdern, J., A problem in additive number theory. Math. Proc. Cambridge Philos. Soc. 103 (1988), 27–33.CrossRefGoogle Scholar
[8]Brüdern, J. and Wooley, T. D., The addition of binary cubic forms. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 356 (1998), 701–737.CrossRefGoogle Scholar
[9]Chowla, S. and Davenport, H., On Weyl’s inequality and Waring’s problem for cubes. Acta Arith. 6 (1961), 505–521.CrossRefGoogle Scholar
[10]Davenport, H., Cubic forms in sixteen variables. Proc. R. Soc. Lond. Ser. A 272 (1963), 285–303.Google Scholar
[11]Davenport, H., Analytic Methods for Diophantine Equations and Diophantine Inequalities, 2nd edn., Cambridge University Press (Cambridge, 2005).CrossRefGoogle Scholar
[12]Dietmann, R., Linear spaces on rational hypersurfaces of odd degree. Bull. Lond. Math. Soc. 42 (2010), 891–895.CrossRefGoogle Scholar
[13]Ford, K. B., New estimates for mean values of Weyl sums. Int. Math. Res. Not. IMRN (1995), 155–171.CrossRefGoogle Scholar
[14]Hartshorne, R., Algebraic Geometry, Springer (Berlin, 1977).CrossRefGoogle Scholar
[15]Heath-Brown, D. R., Cubic forms in ten variables. Proc. Lond. Math. Soc. (3) 47 (1983), 225–257.CrossRefGoogle Scholar
[16]Heath-Brown, D. R., Weyl’s inequality, Hua’s inequality, and Waring’s problem. J. Lond. Math. Soc. (2) 38 (1988), 216–230.CrossRefGoogle Scholar
[17]Heath-Brown, D. R., Cubic forms in 14 variables. Invent. Math. 170 (2007), 199–230.CrossRefGoogle Scholar
[18]Hooley, C., On nonary cubic forms. J. Reine Angew. Math. 386 (1988), 32–98.Google Scholar
[19]Hua, L.-K., An improvement of Vinogradov’s mean-value theorem and several applications. Q. J. Math. 20 (1949), 48–61.CrossRefGoogle Scholar
[20]Hua, L.-K., Additive Theory of Prime Numbers, American Mathematical Society (Providence, RI, 1965).Google Scholar
[21]Lewis, D. J. and Schulze-Pillot, R., Linear spaces on the intersection of cubic hypersurfaces. Monatsh. Math. 97 (1984), 277–285.CrossRefGoogle Scholar
[22]Parsell, S. T., The density of rational lines on cubic hypersurfaces. Trans. Amer. Math. Soc. 352 (2000), 5045–5062.CrossRefGoogle Scholar
[23]Parsell, S. T., Multiple exponential sums over smooth numbers. J. Reine Angew. Math. 532 (2001), 47–104.Google Scholar
[24]Parsell, S. T., A generalization of Vinogradov’s mean value theorem. Proc. Lond. Math. Soc. (3) 91 (2005), 1–32.CrossRefGoogle Scholar
[25]Parsell, S. T., Asymptotic estimates for rational linear spaces on hypersurfaces. Trans. Amer. Math. Soc. 361 (2009), 2929–2957.CrossRefGoogle Scholar
[26]Schmidt, W. M., The density of integer points on homogeneous varieties. Acta. Math. 154 (1985), 243–296.CrossRefGoogle Scholar
[27]Vaughan, R. C., On Waring’s problem for cubes. J. Reine Angew. Math. 365 (1986), 122–170.Google Scholar
[28]Vaughan, R. C., On Waring’s problem for smaller exponents, II. Mathematika 33 (1986), 6–22.CrossRefGoogle Scholar
[29]Vaughan, R. C., A new iterative method in Waring’s problem. Acta Math. 162 (1989), 1–71.CrossRefGoogle Scholar
[30]Vaughan, R. C., The Hardy–Littlewood Method, 2nd edn., Cambridge University Press (Cambridge, 1997).CrossRefGoogle Scholar
[31]Vaughan, R. C. and Wooley, T. D., Further improvements in Waring’s problem, IV: Higher powers. Acta Arith. 94 (2000), 203–285.CrossRefGoogle Scholar
[32]Vinogradov, I. M., New estimates for Weyl sums. Dokl. Akad. Nauk SSSR 8 (1935), 195–198.Google Scholar
[33]Wooley, T. D., Large improvements in Waring’s problem. Ann. of Math. (2) 135 (1992), 131–164.CrossRefGoogle Scholar
[34]Wooley, T. D., On Vinogradov’s mean value theorem. Mathematika 39 (1992), 379–399.CrossRefGoogle Scholar
[35]Wooley, T. D., Linear spaces on cubic hypersurfaces and pairs of homogeneous cubic equations. Bull. Lond. Math. Soc. 29 (1997), 556–562.CrossRefGoogle Scholar
[36]Wooley, T. D., On exponential sums over smooth numbers. J. Reine Angew. Math. 488 (1997), 79–140.Google Scholar
[37]Wooley, T. D., An explicit version of Birch’s theorem. Acta Arith. 85 (1998), 79–96.CrossRefGoogle Scholar
[38]Wooley, T. D., On Weyl’s inequality, Hua’s lemma, and exponential sums over binary forms. Duke Math. J. 100 (1999), 373–423.CrossRefGoogle Scholar
[39]Wooley, T. D., The asymptotic formula in Waring’s problem, Int. Math. Res. Not. IMRN (in press).Google Scholar
[40]Wooley, T. D., Vinogradov’s mean value theorem via efficient congruencing, Ann. of Math. (2) (in press).Google Scholar