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Asymptotic Formulae for Pairs of Diagonal Cubic Equations

Published online by Cambridge University Press:  20 November 2018

Jörg Brüdern  and
Trevor D. Wooley
Jörg Brüdern
Affiliation:
Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstrasse 3-5, D-37073 Göttingen, Germany email: [email protected]
Trevor D. Wooley
Affiliation:
School of Mathematics, University of Bristol, University Walk, Clifton, Bristol BS8 1TW, United Kingdom email: [email protected]
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Abstract

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We investigate the number of integral solutions possessed by a pair of diagonal cubic equations in a large box. Provided that the number of variables in the system is at least fourteen, and in addition the number of variables in any non-trivial linear combination of the underlying forms is at least eight, we obtain an asymptotic formula for the number of integral solutions consistent with the product of local densities associated with the system.

Keywords

11D7211P55exponential sumsDiophantine equations
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 1 , 01 February 2011 , pp. 38 - 54
Copyright
Copyright © Canadian Mathematical Society 2011

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] Boklan, K. D., A reduction technique in Waring's problem, I. Acta Arith. 65(1993), no. 2, 147-161.Google Scholar
[3] Brüdern, J., On pairs of diagonal cubic forms. Proc. London Math. Soc. (3) 61(1990), no. 2, 273-343. doi:10.1112/plms/s3-61.2.273Google Scholar
[4] Brüdern, J. and D.Wooley, T., Hua's lemma and simultaneous diagonal equations. Bull. London Math. Soc. 34(2002), no. 3. 279-283. doi:10.1112/S002460930100889XGoogle Scholar
[5] Brüdern, J. and D.Wooley, T., The paucity problem for certain pairs of diagonal equations. Q. J. Math. 54(2003), no. 1, 41-48. doi:10.1093/qjmath/54.1.41Google Scholar
[6] Brüdern, J. and D.Wooley, T., Asymptotic formulae for pairs of diagonal equations. Math. Proc. Cambridge Philos. Soc. 137(2004), no. 1, 227-235. doi:10.1017/S0305004103007400Google Scholar
[7] Brüdern, J. and D.Wooley, T., The density of integral solutions for pairs of diagonal cubic equations. In: Analytic number theory, Clay Math. Proc., 7, American Mathematical Society, Providence, RI, 2007, pp. 57-76.Google Scholar
[8] Brüdern, J. and D.Wooley, T., The Hasse principle for pairs of diagonal cubic forms. Ann. of Math. 166(2007), no. 3, 865-895. doi:10.4007/annals.2007.166.865Google Scholar
[9] Cook, R. J., Pairs of additive equations. Michigan Math. J. 19(1972), 325-331. doi:10.1307/mmj/1029000942Google Scholar
[10] Cook, R. J., Pairs of additive congruences: cubic congruences. Mathematika 32(1985), no. 2, 286-300. doi:10.1112/S0025579300011062Google Scholar
[11] Davenport, H. and Lewis, D. J., Cubic equations of additive type. Philos. Trans. Roy. Soc. London Ser. A 261(1966), 97-136. doi:10.1098/rsta.1966.0060Google Scholar
[12] Hall, R. R. and Tenenbaum, G., Divisors. Cambridge Tracts in Mathematics, 90, Cambridge University Press, Cambridge, 1988.Google Scholar
[13] Heath-Brown, D. R., The density of rational points on cubic surfaces. Acta Arith. 79(1997), no. 1, 17-30.Google Scholar
[14] Hooley, C., On the numbers that are representable as the sum of two cubes. J. Reine Angew. Math. 314(1980), 146-173.Google Scholar
[15] Vaughan, R. C., On pairs of additive cubic equations. Proc. London Math. Soc. (3) 34(1977), no. 2, 354-364. doi:10.1112/plms/s3-34.2.354Google Scholar
[16] Vaughan, R. C., On Waring's problem for cubes. J. Reine Angew. Math. 365(1986), 122-170.Google Scholar
[17] Vaughan, R. C., A new iterative method in Waring's problem. Acta Math. 162(1989), no. 1-2, 1-71. doi:10.1007/BF02392834Google Scholar
[18] Vaughan, R. C., The Hardy-Littlewood method. Second ed., Cambridge Tracts in Mathematics, 125, Cambridge University Press, Cambridge, 1997.Google Scholar
[19] Wooley, T. D., Slim exceptional sets for sums of cubes. Canad. J. Math. 54(2002), no. 2, 417-448.Google Scholar