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Study of rational cubic forms via the circle method [after D. R. Heath- Brown, C. Hooley, and R. C. Vaughan]. (English) Zbl 0716.11047

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 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 D. R. Heath-Brown [Proc. Lond. Math. Soc., III. Ser. 47, 225-257 (1983; Zbl 0494.10012)] and 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.

[For the original French text see Sémin. Bourbaki Exp. 16. No.720].

Reviewer: J.Brüdern

11P55Applications of the Hardy-Littlewood method
11P05Waring’s problem and variants
11D25Cubic and quartic diophantine equations
11-02Research monographs (number theory)
11D85Representation problems of integers
11D72Equations in many variables