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Rational points on linear slices of diagonal hypersurfaces. (English) Zbl 1371.11139

This paper provides an asymptotic formula for the number of integer solutions to the simultaneous Diophantine equations \[ \sum_{i=1}^s a_i x_i^k=\sum_{i=1}^s b_i x_i=0, \] lying in a large box \(\max|x_i|\leq P\). One supposes that the coefficients are integers and that none of the \(a_i\) vanish. Moreover one assumes that the equations have a nonsingular solution in every completion of \(\mathbb{Q}\). Then if \(k\geq 3\) and \(s\geq 2^k+2\) the number of solutions is asymptotically \(cP^{s-k-1}\) as \(P\to\infty\), for some positive constant \(c\), which may depend on the coefficients \(a_i\) and \(b_i\).
The key tool is a 2-variable mean value estimate for \[ S(\alpha\beta)=\sum_{n\leq P}e(\alpha n^k+\beta n), \] stating that \[ \int_0^1\int_{\mathfrak{m}}|f(\alpha,\beta)|^{2^k}\,d\alpha\, d\beta\ll P^{2^k-k+1}(\log P)^{-2}, \] for a suitable set \(\mathfrak{m}\) of minor arcs. This is proved following the methods of R. C. Vaughan [Mathematika 33, 6–22 (1986; Zbl 0601.10037)], but using a paucity result for the system \[ x_1^k+x_2^k+x_3^k=y_1^k+y_2^k+y_3^k,\;\;\; x_1+x_2+x_3=y_1+y_2+y_3 \] at a suitable point.
As the authors observe, the main interest in their results is for relatively small values of \(k\), since one can now handle values of \(s\) from around \(k^2\) onward, using estimates for Vinogradov’s mean value due to J. Bourgain et al. [Ann. Math. (2) 184, No. 2, 633–682 (2016; Zbl 1408.11083)].

MSC:

11P55 Applications of the Hardy-Littlewood method
11D45 Counting solutions of Diophantine equations
11L15 Weyl sums
14G05 Rational points
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References:

[1] B. J. Birch, Forms in many variables , Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 265 (1961/1962), 245-263. · Zbl 0103.03102 · doi:10.1098/rspa.1962.0007
[2] K. D. Boklan, A reduction technique in Waring’s problem , I , Acta Arith. 65 (1993), 147-161. · Zbl 0785.11049
[3] R. de la Bretèche, Répartition des points rationnels sur la cubique de Segre , Proc. Lond. Math. Soc. (3) 95 (2007), 69-155. · Zbl 1126.14025 · doi:10.1112/plms/pdm001
[4] T. D. Browning and D. R. Heath-Brown, Rational points on quartic hypersurfaces , J. Reine Angew. Math. 629 (2009), 37-88. · Zbl 1169.11027 · doi:10.1515/CRELLE.2009.026
[5] J. Brüdern, A problem in additive number theory , Math. Proc. Cambridge Philos. Soc. 103 (1988), 27-33. · Zbl 0655.10041 · doi:10.1017/S0305004100064586
[6] J. Brüdern, and R. J. Cook, On simultaneous diagonal equations and inequalities , Acta Arith. 62 (1992), 125-149. · Zbl 0774.11015
[7] J. W. S. Cassels and M. J. T. Guy, On the Hasse principle for cubic surfaces , Mathematika 13 (1966), 111-120. · Zbl 0151.03405 · doi:10.1112/S0025579300003879
[8] J. H. H. Chalk, On Hua’s estimates for exponential sums , Mathematika 34 (1987), 115-123. · Zbl 0621.10024 · doi:10.1112/S002557930001336X
[9] H. Davenport and D. J. Lewis, Cubic equations of additive type , Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 261 (1966), 97-136. · Zbl 0227.10038 · doi:10.1098/rsta.1966.0060
[10] H. Davenport and D.J. Lewis, Simultaneous equations of additive type , Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 264 (1969), 557-595. · Zbl 0207.35304 · doi:10.1098/rsta.1969.0035
[11] G. Greaves, Some Diophantine equations with almost all solutions trivial , Mathematika 44 (1997), 14-36. · Zbl 0880.11031 · doi:10.1112/S002557930001192X
[12] H. Halberstam and H.-E. Richert, Sieve Methods , London Math. Soc. Monogr. Ser. 4 , Academic Press, London, 1974. · Zbl 0298.10026
[13] R. R. Hall and G. Tenenbaum, Divisors , Cambridge Tracts in Math. 90 , Cambridge University Press, Cambridge, 1988.
[14] M. P. Harvey, Minor arc moments of Weyl sums , Glasg. Math. J. 55 (2013), 97-113. · Zbl 1309.11071 · doi:10.1017/S0017089512000365
[15] C. Hooley, On the representation of a number as the sum of two \(h\)-th powers , Math. Z. 84 (1964), 126-136. · Zbl 0123.04303 · doi:10.1007/BF01117120
[16] C. Hooley, On a new technique and its applications to the theory of numbers , Proc. Lond. Math. Soc. (3) 38 (1979), 115-151. · Zbl 0394.10027 · doi:10.1112/plms/s3-38.1.115
[17] C. Hooley, On another sieve method and the numbers that are a sum of two \(h\)th powers , Proc. Lond. Math. Soc. (3) 43 (1981), 73-109. · Zbl 0463.10035 · doi:10.1112/plms/s3-43.1.73
[18] C. Hooley, On nonary cubic forms , J. Reine Angew. Math. 386 (1988), 32-98. · Zbl 0641.10019 · doi:10.1515/crll.1988.386.32
[19] C. Hooley, On nonary cubic forms, II , J. Reine Angew. Math. 415 (1991), 95-165. · Zbl 0719.11017 · doi:10.1515/crll.1991.415.95
[20] C. Hooley, On nonary cubic forms, III , J. Reine Angew. Math. 456 (1994), 53-63. · Zbl 0833.11048 · doi:10.1515/crll.1994.456.53
[21] C. Hooley, On another sieve method and the numbers that are a sum of two \(h\)th powers, II , J. Reine Angew. Math. 475 (1996), 55-75. · Zbl 0848.11041 · doi:10.1515/crll.1996.475.55
[22] L. K. Hua, Additive Theory of Prime Numbers , Transl. Math. Monogr. 13 , Amer. Math. Soc., Providence, 1965. · Zbl 0192.39304
[23] S. T. Parsell, Pairs of additive equations of small degree , Acta Arith. 104 (2002), 345-402. · Zbl 1004.11055 · doi:10.4064/aa104-4-2
[24] C. M. Skinner and T. D. Wooley, Sums of two \(k\)th powers , J. Reine Angew. Math. 462 (1995), 57-68. · Zbl 0820.11059
[25] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function , 2nd ed., Clarendon Press, Oxford University Press, New York, 1986. · Zbl 0601.10026
[26] R. C. Vaughan, On Waring’s problem for cubes , J. Reine Angew. Math. 365 (1986), 122-170. · Zbl 0574.10046 · doi:10.1515/crll.1986.365.122
[27] R.C. Vaughan, On Waring’s problem for smaller exponents, II , Mathematika 33 (1986), 6-22. · Zbl 0601.10037 · doi:10.1112/S0025579300013838
[28] R.C. Vaughan, A new iterative method in Waring’s problem , Acta Math. 162 (1989), 1-71. · Zbl 0665.10033 · doi:10.1007/BF02392834
[29] R. C. Vaughan, The Hardy-Littlewood Method , 2nd ed., Cambridge Tracts in Math. 125 , Cambridge University Press, Cambridge, 1997. · Zbl 0868.11046
[30] R.C. Vaughan, “On generating functions in additive number theory, I” in Analytic Number Theory , Cambridge University Press, Cambridge, 2009, 436-448. · Zbl 1221.11174
[31] R. C. Vaughan and T. D. Wooley, On a certain nonary cubic form and related equations , Duke Math. J. 80 (1995), 669-735. · Zbl 0847.11052 · doi:10.1215/S0012-7094-95-08023-5
[32] T. D. Wooley, On simultaneous additive equations, II , J. Reine Angew. Math. 419 (1991), 141-198.
[33] T.D. Wooley, On simultaneous additive equations, III , Mathematika 37 (1990), 85-96. · Zbl 0691.10008 · doi:10.1112/S0025579300012821
[34] T.D. Wooley, Sums of two cubes , Int. Math. Res. Not. IMRN 1995 , no. 4, 181-184. · Zbl 0821.11049 · doi:10.1155/S1073792895000146
[35] T.D. Wooley, The asymptotic formula in Waring’s problem , Int. Math. Res. Not. IMRN 2012 , no. 7, 1485-1504. · Zbl 1267.11104 · doi:10.1093/imrn/rnr074
[36] T.D. Wooley, Vinogradov’s mean value theorem via efficient congruencing , Ann. of Math. (2) 175 (2012), 1575-1627. · Zbl 1267.11105 · doi:10.4007/annals.2012.175.3.12
[37] T.D. Wooley, Vinogradov’s mean value theorem via efficient congruencing, II , Duke Math. J. 162 (2013), 673-730. · Zbl 1312.11066 · doi:10.1215/00127094-2079905
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