The density of rational lines on cubic hypersurfaces

Parsell, Scott

doi:10.1090/S0002-9947-00-02635-0

The density of rational lines on cubic hypersurfaces
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by Scott T. Parsell PDF

Trans. Amer. Math. Soc. 352 (2000), 5045-5062 Request permission

Abstract:

We provide a lower bound for the density of rational lines on the hypersurface defined by an additive cubic equation in at least $57$ variables. In the process, we obtain a result on the paucity of non-trivial solutions to an associated system of Diophantine equations.

References

G. I. Arhipov, A. A. Karacuba, and V. N. Čubarikov, Multiple trigonometric sums, Trudy Mat. Inst. Steklov. 151 (1980), 128 (Russian). MR 608411
R. C. Baker, Diagonal cubic equations. II, Acta Arith. 53 (1989), no. 3, 217–250. MR 1032826, DOI 10.4064/aa-53-3-217-250
B. J. Birch, Homogeneous forms of odd degree in a large number of variables, Mathematika 4 (1957), 102–105. MR 97359, DOI 10.1112/S0025579300001145
P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
T. Estermann, Einige Sätze über quadratfreie Zahlen, Math. Ann. 105 (1931), 653–662.
D. R. Heath-Brown, The density of rational points on cubic surfaces, Acta Arith. 79 (1997), no. 1, 17–30. MR 1438113, DOI 10.4064/aa-79-1-17-30
C. Hooley, On the representations of a number as the sum of four cubes. I, Proc. London Math. Soc. (3) 36 (1978), no. 1, 117–140. MR 506025, DOI 10.1112/plms/s3-36.1.117
Christopher Hooley, On the numbers that are representable as the sum of two cubes, J. Reine Angew. Math. 314 (1980), 146–173. MR 555910, DOI 10.1515/crll.1980.314.146
M. Jurchescu, On analytic maps of analytic spaces, Rev. Roumaine Math. Pures Appl. 9 (1964), 253–264. MR 185144
S. T. Parsell, Multiple exponential sums over smooth numbers J. Reine Angew. Math (to appear).
Wolfgang M. Schmidt, On cubic polynomials. III. Systems of $p$-adic equations, Monatsh. Math. 93 (1982), no. 3, 211–223. MR 661569, DOI 10.1007/BF01299298
Wolfgang M. Schmidt, On cubic polynomials. IV. Systems of rational equations, Monatsh. Math. 93 (1982), no. 4, 329–348. MR 666834, DOI 10.1007/BF01295233
R. C. Vaughan, A new iterative method in Waring’s problem, Acta Math. 162 (1989), no. 1-2, 1–71. MR 981199, DOI 10.1007/BF02392834
R. C. Vaughan, The Hardy-Littlewood method, 2nd ed., Cambridge Tracts in Mathematics, vol. 125, Cambridge University Press, Cambridge, 1997. MR 1435742, DOI 10.1017/CBO9780511470929
Hermann Kober, Transformationen von algebraischem Typ, Ann. of Math. (2) 40 (1939), 549–559 (German). MR 96, DOI 10.2307/1968939
Trevor D. Wooley, On simultaneous additive equations. I, Proc. London Math. Soc. (3) 63 (1991), no. 1, 1–34. MR 1105717, DOI 10.1112/plms/s3-63.1.1
Trevor D. Wooley, Breaking classical convexity in Waring’s problem: sums of cubes and quasi-diagonal behaviour, Invent. Math. 122 (1995), no. 3, 421–451. MR 1359599, DOI 10.1007/BF01231451
Trevor D. Wooley, Sums of two cubes, Internat. Math. Res. Notices 4 (1995), 181–184. MR 1326063, DOI 10.1155/S1073792895000146
Trevor D. Wooley, Linear spaces on cubic hypersurfaces, and pairs of homogeneous cubic equations, Bull. London Math. Soc. 29 (1997), no. 5, 556–562. MR 1458715, DOI 10.1112/S0024609397003184
—, Sums of three cubes, Mathematika (to appear).