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.