The polynomial numerical index of a Banach space

In this paper, we introduce the polynomial numerical index of order k of a Banach space, generalizing to k-homogeneous polynomials the 'classical' numerical index defined by Lumer in the 1970s for linear operators. We also prove some results. Let k be a positive integer. We then have the following: (i) n((k)) (C(K)) = 1 for every scattered compact space K. (ii) The inequality n((k)) (E) >= k(k/(1-k)) for every complex Banach space E and the constant k(k/(1-k)) is sharp. (iii) The inequalities n((k))(E) <= n((k-1))(E) <= k((k+(1/(k-1))))/(k-1)(k-1)n((k))(E) for every Banach space E. (iv) The relation between the polynomial numerical index of c(0), l(1), l(infinity) sums of Banach spaces and the infimum of the polynomial numerical indices of them. (v) The relation between the polynomial numerical index of the space C(K, E) and the polynomial numerical index of E. (vi) The inequality n((k)) (E**) <= n((k)) (E) for every Banach space E. Finally, some results about the numerical radius of multilinear maps and homogeneous polynomials on C(K) and the disc algebra are given.