  We study the computational complexity of a criterion and an algorithm for
diagonalization of multivariate homogeneous polynomials, that is, expressing
them as sums of powers of independent linear forms. They are based on
Harrison's center theory and only require solving linear and quadratic systems
of equations. Detailed descriptions and computational complexity of each step
of the algorithm are provided. The complexity analysis focuses on the impacts
of problem sizes, including the number of variables and the degree of given
polynomials. We show that this algorithm runs in polynomial time and validate
it through numerical experiments. Other diagonalization algorithms are reviewed
and compared in terms of complexity.
