Power sum polynomials and the ghosts behind them

The power sum polynomial associated to a multi-subset of the projective plane PG(2,q) is the sum of the (q−1)-th powers of the Rédei factors of the points in the multi-subset. The classification of multi-subsets having the same power sum polynomial passes through the determination of those multi-subsets associated to the zero polynomial, called ghosts. In this paper we provide new classes of ghosts and compute the dimension of the ghost subspace by exploiting the linear code generated by the lines of PG(2,q) and its dual. Moreover, we explicitly enumerate and classify ghosts for planes of order 2, 3, 4.