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Zbl 0716.11047
Deshouillers, Jean-Marc
Study of rational cubic forms via the circle method [after D. R. Heath- Brown, C. Hooley, and R. C. Vaughan]. (English)
[J] Sémin. Théor. Nombres Bordx., Sér. II 2, No.2, 431-450 (1990). ISSN 0989-5558

This paper surveys the development of the circle method from Hardy and Littlewood to its recent renaissance, with special emphasis on cubic forms. The author is thus concerned with Waring's problem for cubes, mentions {\it R. C. Vaughan's} asymptotic formula for 8 cubes and the lower bound for 7 cubes [J. Reine Angew. Math. 365, 122-170 (1986; Zbl 0574.10046), J. Lond. Math. Soc., II. Ser. 39, 205-218 (1989; Zbl 0677.10034)] and then turns to nondiagonal problems. Here the climax so far are the achievements of {\it D. R. Heath-Brown} [Proc. Lond. Math. Soc., III. Ser. 47, 225-257 (1983; Zbl 0494.10012)] and {\it C. Hooley} [J. Reine Angew. Math. 386, 32-98 (1988; Zbl 0641.10019)]. Some topics within this circle of ideas are omitted, in particular the work on the four cubes problem initiated by Davenport. \par [For the original French text see Sémin. Bourbaki Exp. 16. No.720].
[J.Brüdern]

MSC 2000:: *11P55 Appl. of the Hardy-Littlewood method
11P05 Waring's problem and variants
11D25 Cubic and quartic diophantine equations
11-02 Research monographs (number theory)
11D85 Representation problems of integers
11D72 Equations in many variables

Keywords: applications of Hardy-Littlewood circle method; cubic forms; Waring's problem for cubes

Citations: Zbl 0574.10046; Zbl 0677.10034; Zbl 0494.10012; Zbl 0641.10019

Cited in: Zbl 0726.11062

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