anti-n-order polynomial daugavet property on banach spaces

We generalize the notion of the anti-daugavet property (a-dp) to the anti-n-order polynomial daugavet property (a-npdp) for banach spaces by identifying a good spectrum of a polynomial and prove that locally uniformly alternatively convex or smooth banach spaces have the a-mdp for rank-1 polynomials. we then prove that locally uniformly convex banach spaces have the a-npdp for compact polynomials if and only if their norms are eigenvalues, and uniformly convex banach spaces have the a-npdp for continuous polynomials if and only if their norms belong to the approximate spectra.