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Rational points on linear slices of diagonal hypersurfaces

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

Published online by Cambridge University Press:  11 January 2016

Jörg Brüdern  and
Olivier Robert
Jörg Brüdern
Affiliation:
Mathematisches Institut, D-37073 Göttingen, Germany, bruedern@uni-math.gwdg.de
Olivier Robert
Affiliation:
Université de Lyon and Université de Saint-Etienne, Institut Camille Jordan CNRS UMR 5208, F-42000 Saint-Etienne, France, olivier.robert@univ-st-etienne.fr
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Abstract

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An asymptotic formula is obtained for the number of rational points of bounded height on the class of varieties described in the title line. The formula is proved via the Hardy-Littlewood method, and along the way we establish two new results on Weyl sums that are of some independent interest.

MSC classification

Secondary: 11D45: Counting solutions of Diophantine equations 11L15: Weyl sums 11P55: Applications of the Hardy-Littlewood method

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 218 , June 2015 , pp. 51 - 100
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

References

[1] Birch, B. J., Forms in many variables, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 265 (1961/1962), 245–263. MR 0150129.Google Scholar
[2] Boklan, K. D., A reduction technique in Waring's problem, I, Acta Arith. 65 (1993), 147–161. MR 1240121.Google Scholar
[3] R. de la Bretèche, , Répartition des points rationnels sur la cubique de Segre, Proc. Lond. Math. Soc. (3) 95(2007), 69–155. MR 2329549. DOI 10.1112/plms/pdm001.Google Scholar
[4] Browning, T. D. and Heath-Brown, D. R., Rational points on quartic hypersurfaces, J. Reine Angew. Math. 629(2009), 37–88. MR 2527413. DOI 10.1515/CRELLE.2009.026.Google Scholar
[5] Brüdern, J., A problem in additive number theory, Math. Proc. Cambridge Philos. Soc. 103(1988), 27–33. MR 0913447. DOI 10.1017/S0305004100064586.Google Scholar
[6] Brüdern, J., and Cook, R. J., On simultaneous diagonal equations and inequalities, Acta Arith. 62(1992), 125–149. MR 1183985.Google Scholar
[7] Cassels, J. W. S. and Guy, M. J. T., On the Hasse principle for cubic surfaces, Math- ematika 13(1966), 111–120. MR 0211966.Google Scholar
[8] Chalk, J. H. H., On Hua's estimates for exponential sums, Mathematika 34(1987), 115–123. MR 0933491. DOI 10.1112/S002557930001336X.Google Scholar
[9] Davenport, H. and Lewis, D. J., Cubic equations of additive type, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 261(1966), 97–136. MR 0205962.Google Scholar
[10] Davenport, H. and Lewis, D. J., Simultaneous equations of additive type, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 264(1969), 557–595. MR 0245542.Google Scholar
[11] Greaves, G., Some Diophantine equations with almost all solutions trivial, Mathematika 44 (1997), 14–36. MR 1464372. DOI 10.1112/S002557930001192X.CrossRefGoogle Scholar
[12] Halberstam, H. and Richert, H.-E., Sieve Methods, London Math. Soc. Monogr. Ser. 4, Academic Press, London, 1974. MR 0424730.Google Scholar
[13] Hall, R. R. and Tenenbaum, G., Divisors, Cambridge Tracts in Math. 90, Cambridge University Press, Cambridge, 1988. MR 0964687. DOI 10.1017/CBO9780511566004. Google Scholar
[14] Harvey, M. P., Minor arc moments of Weyl sums, Glasg. Math. J. 55 (2013), 97–113. MR 3001332. DOI 10.1017/S0017089512000365.Google Scholar
[15] Hooley, C., On the representation of a number as the sum of two h-th powers, Math. Z. 84 (1964), 126–136. MR 0162767.CrossRefGoogle Scholar
[16] Hooley, C., On a new technique and its applications to the theory of numbers, Proc. Lond. Math. Soc. (3) 38 (1979), 115–151. MR 0520975. DOI 10.1112/plms/s3-38.1.115.Google Scholar
[17] Hooley, C., On another sieve method and the numbers that are a sum of two hth powers, Proc. Lond. Math. Soc. (3) 43 (1981), 73–109. MR 0623719. DOI 10.1112/plms/s3-43.1.73.CrossRefGoogle Scholar
[18] Hooley, C., On nonary cubic forms, J. Reine Angew. Math. 386 (1988), 32–98. MR 0936992. DOI 10.1515/crll.1988.386.32.Google Scholar
[19] Hooley, C., On nonary cubic forms, II, J. Reine Angew. Math. 415 (1991), 95–165. MR 1096903. DOI 10.1515/crll.1991.415.95.Google Scholar
[20] Hooley, C., On nonary cubic forms, III, J. Reine Angew. Math. 456 (1994), 53–63. MR 1301451. DOI 10.1515/crll.1994.456.53.Google Scholar
[21] Hooley, C., On another sieve method and the numbers that are a sum of two hth powers, II, J. Reine Angew. Math. 475 (1996), 55–75. MR 1396726. DOI 10.1515/crll.1996.475.55.Google Scholar
[22] Hua, L. K., Additive Theory of Prime Numbers, Transl. Math. Monogr. 13, Amer. Math. Soc., Providence, 1965. MR 0194404.Google Scholar
[23] Parsell, S. T., Pairs of additive equations of small degree, Acta Arith. 104 (2002), 345–402. MR 1911162. DOI 10.4064/aa104-4-2.Google Scholar
[24] Skinner, C. M. and Wooley, T. D., Sums of two kth powers, J. Reine Angew. Math. 462 (1995), 57–68. MR 1329902.Google Scholar
[25] Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, 2nd ed., Clarendon Press, Oxford University Press, New York, 1986. MR 0882550.Google Scholar
[26] Vaughan, R. C., On Waring's problem for cubes, J. Reine Angew. Math. 365 (1986), 122–170. MR 0826156. DOI 10.1515/crll.1986.365.122.Google Scholar
[27] Vaughan, R. C., On Waring's problem for smaller exponents, II, Mathematika 33 (1986), 6–22. MR 0859494. DOI 10.1112/S0025579300013838.Google Scholar
[28] Vaughan, R. C., A new iterative method in Waring's problem, Acta Math. 162 (1989), 1–71. MR 0981199. DOI 10.1007/BF02392834.Google Scholar
[29] Vaughan, R. C., The Hardy-Littlewood Method, 2nd ed., Cambridge Tracts in Math. 125, Cambridge University Press, Cambridge, 1997. MR 1435742. DOI 10.1017/CBO9780511470929.Google Scholar
[30] Vaughan, R. C., “On generating functions in additive number theory, I” in Analytic Number Theory, Cambridge University Press, Cambridge, 2009, 436–448. MR 2508662.Google Scholar
[31] Vaughan, R. C. and Wooley, T. D., On a certain nonary cubic form and related equations, Duke Math. J. 80 (1995), 669–735. MR 1370112. DOI 10.1215/S0012-7094-95-08023-5.Google Scholar
[32] Wooley, T. D., On simultaneous additive equations, II, J. Reine Angew. Math. 419 (1991), 141–198. MR 1116923. DOI 10.1515/crll.1991.419.141.Google Scholar
[33] Wooley, T. D., On simultaneous additive equations, III, Mathematika 37 (1990), 85–96. MR 1067890. DOI 10.1112/S0025579300012821.Google Scholar
[34] Wooley, T. D., Sums of two cubes, Int. Math. Res. Not. IMRN 1995, no. 4, 181–184. MR 1326063. DOI 10.1155/S1073792895000146.Google Scholar
[35] Wooley, T. D., The asymptotic formula in Waring's problem, Int. Math. Res. Not. IMRN 2012, no. 7, 1485–1504.Google Scholar
[36] Wooley, T. D., Vinogradov 's mean value theorem via efficient congruencing, Ann. of Math. (2) 175 (2012), 1575–1627. MR 2912712. DOI 10.4007/annals.2012.175.3.12.Google Scholar
[37] Wooley, T. D., Vinogradov's mean value theorem via efficient congruencing, II, Duke Math. J. 162 (2013), 673–730. MR 3039678. DOI 10.1215/00127094-2079905.Google Scholar