For any n × n nonnegative matrix A, and any norm . on R n , η . (A) is defined as sup { A⊗x x : x ∈ R n + , x = 0}. Let P (λ) be a matrix polynomial in the max algebra. In this paper, we introduce η . [P (λ)], as a generalization of the matrix norm η . (.), and we investigate some algebraic properties of this notion. We also study some properties of the maximum circuit geometric mean of the companion matrix of P (λ) and the relationship between this concept and the matrices P (1) and coefficients of P (λ). Some properties of η . (Ψ), for a bounded set of max matrix polynomials Ψ, are also investigated.