The real rank of monomials

We consider the problem of determining the real rank of a real homogeneous polynomial f of degree d, i.e. the smallest possible number r such that f can be written as a sum of r dth powers of real linear forms. In geometric complexity theory the real rank of a polynomial is considered — among other notions of rank — as a measure of the computational complexity of evaluating the polynomial. After some general preliminaries we will present upper and lower bounds on the real rank of a monomial and we characterize those monomials for which the lower bound is attained. This is joint work with Enrico Carlini, Alessandro Oneto and Emanuele Ventura.