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
1570283293 2048050639 2035417470 2133160967
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 . 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 . 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.