higher numerical ranges of matrix polynomials

let $p(lambda)$ be an $n$-square complex matrix polynomial, and $1 leq k leq n$ be a positive integer. in this paper, some algebraic and geometrical properties of the $k$-numerical range of $p(lambda)$ are investigated. in particular, the relationship between the $k$-numerical range of $p(lambda)$ and the $k$-numerical range of its companion linearization is stated. moreover, the $k$-numerical range of the basic $a$-factor block circulant matrix, which is the block companion matrix of the matrix polynomial $p(lambda) = lambda ^m i_n - a$, is studied.