ON WARING PROBLEM FOR CUBIC FORMS

Let w be a cubic form in n variables over the complex field, and r be the smallest integer such that w can be expressed as a sum of cubes of r linear forms. Some theorems are proved that connect r with the ranks of certain matrices associated with w for arbitrary n and for n = 5. Based on these theorems, an algorithm to express a cubic form in five variables as a sum of cubes of linear forms is proposed. The paper is concluded with two examples of indecomposable cubic forms: for the first form it is shown that r = 7 and that there exists a one-parameter set of representations of the form as a sum of cubes of seven linear forms, while for the second form it is shown that r greater-than-or-equal-to 8.