Transactions of the American Mathematical Society

Author(s): Scott T. Parsell
Journal: Trans. Amer. Math. Soc. 352 (2000), 5045-5062.
MSC (2000): Primary 11D25, 11D45, 11L03, 11P55
Posted: July 18, 2000
Retrieve article in: PDF
This article is available free of charge

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.

1.: G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov, Multiple trigonometric sums, Trudy Mat. Inst. Steklov. 151 (1980); English Transl., Proc. Steklov Inst. Math. 1982, no. 2 (151). MR 82i:10045
2.: R. C. Baker, Diagonal cubic equations II, Acta Arith. 53 (1989), 217-250. MR 91b:11100a
3.: B. J. Birch, Homogeneous forms of odd degree in a large number of variables, Mathematika 4 (1957), 102-105. MR 20:3828
4.: R. Brauer, A note on systems of homogeneous algebraic equations, Bull. Amer. Math. Soc. 51 (1945), 749-755. MR 7:108i
5.: T. Estermann, Einige Sätze über quadratfreie Zahlen, Math. Ann. 105 (1931), 653-662.
6.: D. R. Heath-Brown, The density of rational points on cubic surfaces, Acta Arith. 79 (1997), 17-30. MR 98h:11083
7.: C. Hooley, On the representations of a number as the sum of four cubes, Proc. London Math. Soc. (3) 36 (1978), 117-140. MR 58:21932a
8.: -, On the numbers that are representable as the sum of two cubes, J. Reine Angew. Math. 314 (1980), 146-173. MR 81d:10036
9.: L.-K. Hua, Additive theory of prime numbers, A.M.S., Providence, RI, 1965. MR 32:2614
10.: S. T. Parsell, Multiple exponential sums over smooth numbers J. Reine Angew. Math (to appear).
11.: W. M. Schmidt, On cubic polynomials III. Systems of -adic equations, Monatsh. Math. 93 (1982), 211-223. MR 83m:10062
12.: -, On cubic polynomials IV. Systems of rational equations, Monatsh. Math. 93 (1982), 329-348. MR 83m:10063
13.: R. C. Vaughan, A new iterative method in Waring's problem, Acta Math. 162 (1989), 1-71. MR 90c:11072
14.: -, The Hardy-Littlewood method, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 98a:11133
15.: R. C. Vaughan and T. D. Wooley, Further improvements in Waring's problem, Acta Math. 174 (1995), 147-240. MR 96j:11129
16.: T. D. Wooley, On simultaneous additive equations II, J. Reine Angew. Math. 429 (1991), 141-198. MR 92e:11025b
17.: -, Breaking classical convexity in Waring's problem: Sums of cubes and quasi-diagonal behaviour, Invent. Math. 122 (1995), 421-451. MR 97d:11148
18.: -, Sums of two cubes, Internat. Math. Res. Notices 4 (1995), 181-185. MR 96a:11103
19.: -, Linear spaces on cubic hypersurfaces and pairs of homogeneous cubic equations, Bull. London Math. Soc. 29 (1997), 556-562. MR 99e:11036
20.: -, Sums of three cubes, Mathematika (to appear).

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11D25, 11D45, 11L03, 11P55

Scott T. Parsell
Affiliation: Department of Mathematics, University of Michigan, 525 East University Avenue, Ann Arbor, Michigan 48109-1109
Address at time of publication: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: parsell@alum.mit.edu

DOI: 10.1090/S0002-9947-00-02635-0
PII: S 0002-9947(00)02635-0
Received by editor(s): May 21, 1999
Received by editor(s) in revised form: July 23, 1999
Posted: July 18, 2000
Additional Notes: Research supported in part by NSF grant DMS-9622773 and by a fellowship from the David and Lucile Packard Foundation.
Copyright of article: Copyright 2000, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS