  In this work, linearized multivariate skew polynomials over division rings
are introduced. Such polynomials are right linear over the corresponding
centralizer and generalize linearized polynomial rings over finite fields,
group rings or differential polynomial rings. Their natural evaluation is
connected to the remainder-based evaluation of free multivariate skew
polynomials. It is shown that P-independent sets are those given by right
linearly independent sets when partitioned into conjugacy classes. Hence
finitely generated P-closed sets correspond to lists of finite-dimensional
right vector spaces, extending Lam and Leroy's results on univariate skew
polynomials. It is also shown that products of free multivariate skew
polynomials translate into coordinate-wise compositions of linearized
multivariate skew polynomials, which in turn translate into matrix products
over the corresponding centralizers. Later, linearized multivariate Vandermonde
matrices are introduced, which generalize multivariate Vandermonde, Moore and
Wronskian matrices. The previous results explicitly give their ranks in
general. P-Galois extensions of division rings are then introduced, which
generalize classical (finite) Galois extensions. Three Galois-theoretic results
are generalized to such extensions: Artin's theorem on extension degrees, the
Galois correspondence and Hilbert's Theorem 90.
