On the polynomial numerical index of the real spaces c0, ℓ1 and ℓ∞

We give a lower bound for the polynomial numerical index of order k for real lush spaces. We use this bound to compute the polynomial numerical index of order 2 of the real spaces c0, ℓ1 and ℓ∞. Finally, we present an example of a real Banach space X whose polynomial numerical indices are positive while the ones of its bidual are zero.