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On simultaneous additive equations, IV

Part of: Diophantine equations

Published online by Cambridge University Press:  26 February 2010

Trevor D. Wooley
Trevor D. Wooley
Affiliation:
Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, MI 48109-1109, U.S.A.
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Extract

Recent advances in the theory of exponential sums (see, for example, [6], [7], [8], [12]) have contributed to corresponding progress in our understanding of the solubility of systems of simultaneous additive equations (see, in particular, [1], [2], [3], [4]). In a previous memoir [11] we developed a version of Vaughan's iterative method (see Vaughan [8]) suitable for the analysis of simultaneous additive equations of differing degrees, discussing in detail the solubility of simultaneous cubic and quadratic equations. The mean value estimates derived in [11] are, unfortunately, weaker than might be hoped, owing to the presence of undesirable singular solutions in certain auxiliary systems of congruences. The methods of [12] provide a flexible alternative to Vaughan's iterative method, and, as was apparent even at the time of their initial development at the opening of the present decade, such ideas provide a means of avoiding altogether the aforementioned problematic singular solutions. The systematic development of such an approach having been described recently in [15], in this paper we apply such methods to investigate the solubility of pairs of additive equations, one cubic and one quadratic, thereby improving the main conclusion of [11].

MSC classification

Secondary: 11D72: Equations in many variables
Type
Research Article
Information
Mathematika , Volume 45 , Issue 2 , December 1998 , pp. 319 - 335
Copyright
Copyright © University College London 1998

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References

1.Baker, R. C. and Brüdern, J.. On pairs of additive cubic equations. J. Reine Angew. Math., 391 (1988), 157–180.Google Scholar
2.Brüdern, J.. On pairs of diagonal cubic forms. Proc. London Math. Soc. (3), 61 (1990), 273–343.CrossRefGoogle Scholar
3.Brüdern, J.. On Waring's problem for fifth powers and some related topics. Proc. London Math. Soc. (3), 61 (1990), 457–479.CrossRefGoogle Scholar
4.Brüdern, J. and Cook, R. J.. On simultaneous diagonal equations and inequalities. Ada Arith., 62 (1992), 125–149.CrossRefGoogle Scholar
5.Hua, L.-K.. Additive Theory of Prime Numbers (American Mathematical Society, Providence, Rhode Island, 1965).Google Scholar
6.Vaughan, R. C.. On Waring's problem for cubes. J. Reine Angew. Math., 365 (1986), 122170.Google Scholar
7.Vaughan, R. C.. On Waring's problem for smaller exponents. Proc. London Math. Soc. (3), 52 (1986), 445–463.CrossRefGoogle Scholar
8.Vaughan, R. C.. A new iterative method in Waring's problem. Ada Math., 162 (1989), 1–71.Google Scholar
9.Vaughan, R. C.. The Hardy-Littlewood Method, 2nd Edition, Cambridge Tract No. 125 (Cambridge University Press, Cambridge, 1997).CrossRefGoogle Scholar
10.Wooley, T. D.. On simultaneous additive equations, I. Proc. London Math. Soc. (3), 63 (1991), 1–34.CrossRefGoogle Scholar
11.Wooley, T. D.. On simultaneous additive equations, II. J. Reine Angew. Math., 419 (1991), 141–198.Google Scholar
12.Wooley, T. D.. Large improvements in Waring's problem. Ann. of Math., 135 (1992), 131164.CrossRefGoogle Scholar
13.Wooley, T. D.. A note on symmetric diagonal equations. Number Theory with an emphasis on the Markoff Spectrum. Pollington, A. D. and Moran, W., eds. (Marcel Dekker, 1993), 317–321.Google Scholar
14.Wooley, T. D.. An affine slicing approach to certain paucity problems. Analytic Number Theory: Proceedings of a Conference in Honor of Heini Halberstam. Berndt, B. C., Diamond, H. G. and Hildebrand, A. J., ed., vol. 2 (Birkhauser Boston, Boston, MA, 1996), 803–815.Google Scholar
15.Wooley, T. D.. On exponential sums over smooth numbers. J. Reine Angew. Math., 488 (1997), 79–140.Google Scholar