The Daugavet equation for polynomials

We study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space X, i.e. when the equality parallel to Id+P parallel to=11+parallel to P parallel to is satisfied for all weakly compact polynomials P: X -> X. We show that this is the case when X = C(K), the real or complex space of continuous functions on a compact space K without isolated points. We also study the alternative Daugavet equation max parallel to I+omega P parallel to=1+parallel to P parallel to vertical bar w vertical bar=1 for polynomials P : X -> X. We show that this equation holds for every polynomial on the complex space X = C(K) (K arbitrary) with values in X. This result is not true in the real case. Finally, we study the Daugavet and the alternative Daugavet equations for k-homogeneous polynomials.