Kleene chain completeness and fixedpoint properties

Let L denote the nonscalar complexity in k(x₁,…, x_n). We prove $\text{L(?,??/?x}{}_1\text{,…,??/?x}{}_{\mathrm{n}}\text{)?3L(?)}$. Using this we determine the complexity of single power sums, single elementary symmetric functions, the resultant and the discriminant as root functions, up to order of magnitude. Also we linearly reduce matrix inversion to computing the determinant.