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Rational points on linear slices of diagonal hypersurfaces

  • Jörg Brüdern (a1) and Olivier Robert (a2)

Abstract

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.

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[1] Birch, B. J., Forms in many variables, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 265 (1961/1962), 245263. MR 0150129.
[2] Boklan, K. D., A reduction technique in Waring's problem, I, Acta Arith. 65 (1993), 147161. MR 1240121.
[3] R. de la Bretèche, , Répartition des points rationnels sur la cubique de Segre, Proc. Lond. Math. Soc. (3) 95(2007), 69155. MR 2329549. DOI 10.1112/plms/pdm001.
[4] Browning, T. D. and Heath-Brown, D. R., Rational points on quartic hypersurfaces, J. Reine Angew. Math. 629(2009), 3788. MR 2527413. DOI 10.1515/CRELLE.2009.026.
[5] Brüdern, J., A problem in additive number theory, Math. Proc. Cambridge Philos. Soc. 103(1988), 2733. MR 0913447. DOI 10.1017/S0305004100064586.
[6] Brüdern, J., and Cook, R. J., On simultaneous diagonal equations and inequalities, Acta Arith. 62(1992), 125149. MR 1183985.
[7] Cassels, J. W. S. and Guy, M. J. T., On the Hasse principle for cubic surfaces, Math- ematika 13(1966), 111120. MR 0211966.
[8] Chalk, J. H. H., On Hua's estimates for exponential sums, Mathematika 34(1987), 115123. MR 0933491. DOI 10.1112/S002557930001336X.
[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), 97136. MR 0205962.
[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), 557595. MR 0245542.
[11] Greaves, G., Some Diophantine equations with almost all solutions trivial, Mathematika 44 (1997), 1436. MR 1464372. DOI 10.1112/S002557930001192X.
[12] Halberstam, H. and Richert, H.-E., Sieve Methods, London Math. Soc. Monogr. Ser. 4, Academic Press, London, 1974. MR 0424730.
[13] Hall, R. R. and Tenenbaum, G., Divisors, Cambridge Tracts in Math. 90, Cambridge University Press, Cambridge, 1988. MR 0964687. DOI 10.1017/CBO9780511566004.
[14] Harvey, M. P., Minor arc moments of Weyl sums, Glasg. Math. J. 55 (2013), 97113. MR 3001332. DOI 10.1017/S0017089512000365.
[15] Hooley, C., On the representation of a number as the sum of two h-th powers, Math. Z. 84 (1964), 126136. MR 0162767.
[16] Hooley, C., On a new technique and its applications to the theory of numbers, Proc. Lond. Math. Soc. (3) 38 (1979), 115151. MR 0520975. DOI 10.1112/plms/s3-38.1.115.
[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), 73109. MR 0623719. DOI 10.1112/plms/s3-43.1.73.
[18] Hooley, C., On nonary cubic forms, J. Reine Angew. Math. 386 (1988), 3298. MR 0936992. DOI 10.1515/crll.1988.386.32.
[19] Hooley, C., On nonary cubic forms, II, J. Reine Angew. Math. 415 (1991), 95165. MR 1096903. DOI 10.1515/crll.1991.415.95.
[20] Hooley, C., On nonary cubic forms, III, J. Reine Angew. Math. 456 (1994), 5363. MR 1301451. DOI 10.1515/crll.1994.456.53.
[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), 5575. MR 1396726. DOI 10.1515/crll.1996.475.55.
[22] Hua, L. K., Additive Theory of Prime Numbers, Transl. Math. Monogr. 13, Amer. Math. Soc., Providence, 1965. MR 0194404.
[23] Parsell, S. T., Pairs of additive equations of small degree, Acta Arith. 104 (2002), 345402. MR 1911162. DOI 10.4064/aa104-4-2.
[24] Skinner, C. M. and Wooley, T. D., Sums of two kth powers, J. Reine Angew. Math. 462 (1995), 5768. MR 1329902.
[25] Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, 2nd ed., Clarendon Press, Oxford University Press, New York, 1986. MR 0882550.
[26] Vaughan, R. C., On Waring's problem for cubes, J. Reine Angew. Math. 365 (1986), 122170. MR 0826156. DOI 10.1515/crll.1986.365.122.
[27] Vaughan, R. C., On Waring's problem for smaller exponents, II, Mathematika 33 (1986), 622. MR 0859494. DOI 10.1112/S0025579300013838.
[28] Vaughan, R. C., A new iterative method in Waring's problem, Acta Math. 162 (1989), 171. MR 0981199. DOI 10.1007/BF02392834.
[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.
[30] Vaughan, R. C., “On generating functions in additive number theory, I” in Analytic Number Theory, Cambridge University Press, Cambridge, 2009, 436448. MR 2508662.
[31] Vaughan, R. C. and Wooley, T. D., On a certain nonary cubic form and related equations, Duke Math. J. 80 (1995), 669735. MR 1370112. DOI 10.1215/S0012-7094-95-08023-5.
[32] Wooley, T. D., On simultaneous additive equations, II, J. Reine Angew. Math. 419 (1991), 141198. MR 1116923. DOI 10.1515/crll.1991.419.141.
[33] Wooley, T. D., On simultaneous additive equations, III, Mathematika 37 (1990), 8596. MR 1067890. DOI 10.1112/S0025579300012821.
[34] Wooley, T. D., Sums of two cubes, Int. Math. Res. Not. IMRN 1995, no. 4, 181184. MR 1326063. DOI 10.1155/S1073792895000146.
[35] Wooley, T. D., The asymptotic formula in Waring's problem, Int. Math. Res. Not. IMRN 2012, no. 7, 14851504.
[36] Wooley, T. D., Vinogradov 's mean value theorem via efficient congruencing, Ann. of Math. (2) 175 (2012), 15751627. MR 2912712. DOI 10.4007/annals.2012.175.3.12.
[37] Wooley, T. D., Vinogradov's mean value theorem via efficient congruencing, II, Duke Math. J. 162 (2013), 673730. MR 3039678. DOI 10.1215/00127094-2079905.
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Rational points on linear slices of diagonal hypersurfaces

  • Jörg Brüdern (a1) and Olivier Robert (a2)

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