Norm or numerical radius attaining polynomials on C(K)

Let C(K, ℂ) be the Banach space of all complex-valued continuous functions on a compact Hausdorff space K. We study when the following statement holds: every norm attaining n-homogeneous complex polynomial on C(K, ℂ) attains its norm at extreme points. We prove that this property is true whenever K is a compact Hausdorff space of dimension less than or equal to one. In the case of a compact metric space a characterization is obtained. As a consequence we show that, for a scattered compact Hausdorff space K, every continuous n-homogeneous complex polynomial on C(K, ℂ) can be approximated by norm attaining ones at extreme points and also that the set of all extreme points of the unit ball of C(K, ℂ) is a norming set for every continuous complex polynomial. Similar results can be obtained if "norm" is replaced by "numerical radius."