On the rank of a symmetric form

Abstract

We give a lower bound for the degree of a finite apolar subscheme of a symmetric form F, in terms of the degrees of the generators of the annihilator ideal of F. In the special case, when F is a monomial x_0^d_0 x_2^d_2... x_n^d_n with d_0<= d_1<=...<=d_n-1<= d_n we deduce that the minimal length of an apolar subscheme of F is (d_0+1)...(d_n-1+1), and if d_0=..=d_n, then this minimal length coincides with the rank of F.