Theory and applications of linearized multivariate skew polynomials

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