Manin’s conjecture for a class of singular cubic hypersurfaces

Abstract

Let l > 2 be a fixed positive integer and Q(y) be a positive definite quadratic form in l variables with integral coefficients. The aim of this paper is to count rational points of bounded height on the cubic hypersurface defined by u³ = Q(y)z. We can get a power-saving result for a class of special quadratic forms and improve on some previous work.

Key words： Manin’s conjecture; quadratic form; asymptotic formula; divisor problem; exponential sum

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