Around the Waring problem for polynomials

We discuss an analog of the classical Waring problem for C[x_0,x_1,...,x_n]. Namely, we show that a general homogeneous polynomial p contained in C[x_0,x_1,...,x_n] of degree divisible by k > 1 can be represented as a sum of at most kn k-th powers of homogeneous polynomials in C[x_0,x_1,...,x_n]. Notably, kn asymptotically coincides with the number obtained by the naive dimension count.