We consider a system of @math cubic forms in @math variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided @math , we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish. In particular we can handle systems of forms in @math variables, previous work having required that @math . One conjectures that @math should be sufficient. We reduce the problem to an upper bound for the number of solutions to a certain auxiliary inequality. To prove this bound we adapt a method of Davenport.
Br "udern and Wooley @cite_0 @cite_7 @cite_13 treat diagonal systems in @math variables. This is the best value of @math possible with the classical circle method. In particular they prove the Hasse principle for @math whenever the @math are diagonal, @math is smooth and @math . They also prove an asymptotic formula of the type whenever @math holds, or when @math and @math holds @cite_11 @cite_8 @cite_15 . In the case @math they prove a Hasse principle for certain pairs of diagonal cubics in as few as 11 variables @cite_9 .
@cite_7 @cite_8 @cite_9 @cite_0 @cite_15 @cite_13 @cite_11
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Consider a system of diagonal equations satisfying the property that the (fixed) integral coefficient matrix @math contains no singular @math submatrix. A recent paper of the authors [ 3 ] establishes that whenever @math and @math , then the expected asymptotic formula holds for the number @math of integral solutions @math of ( @math ) with @math @math . We investigate pairs of diagonal cubic equations with integral coefficients. For a class of such Diophantine systems with 11 or more variables, we are able to establish that the number of integral solutions in a large box is at least as large as the expected order of magnitude. By means of the Hardy-Littlewood method, we apply a new mean value theorem for exponential sums to confirm the truth, over the rational numbers, of the Hasse principle for pairs of diagonal cubic forms in thirteen or more variables. We establish the Hasse principle for systems of r simultaneous diagonal cubic equations whenever the number of variables exceeds 6r and the associated coefficient matrix contains no singular (r r ) submatrix, thereby achieving the theoretical limit of the circle method for such systems. This paper concerns systems of r homogeneous diagonal equations of degree k in s variables, with integer coefficients. Subject to a suitable non-singularity condition, it is shown that the expected asymptotic formula holds for the number of such systems inside a box [− P , P ] s , provided only that s > (3 r +1)2 k −2 . By way of comparison, classical methods based on the use of Hua's lemma would establish a similar conclusion, provided instead that s > r 2 k .