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We prove an asymptotic formula for the number of representations of squaresby nonsingular cubic forms in six or more variables. The main ingredients ofthe proof are Heath-Brown's form of the Circle Method and various exponentialsum results. The depth of the exponential sum results is comparable to Hooley'swork on cubic forms in nine variables, in particular we prove an analogue ofKatz' bound.

Representation of Squares by Nonsingular Cubic Forms