The main result of this paper is a proof of the expected asymptotic formula for the density of zeros of a family of cubic forms in seven variables. This is established using the Hardy-Littlewood circle method.

Abstract. We obtain an improved bound for the 2 k -th moment of a degree k Weyl sum, restricted to a set of minor arcs, when k is small. We then present some applications of this bound to some Diophantine problems, including a case of the Waring-Goldbach problem, and a particular family of Diophantine equations defined as the sum of a norm form and a diagonal form.2010 Mathematics Subject Classification. 11L15, 11P55.

We apply Freeman's variant of the Davenport–Heilbronn method to provide an asymptotic formula for the number of small values taken by a certain family of cubic forms with real coefficients. The cubic forms in question arise as the sum of a diagonal form and a norm form and should have at least seven variables.

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