Structure and arithmetic of multivariate Ore extensions

We give the basic structure of the multivariable Ore extensions $S=A[\underset{‾}{t};\sigma ,\underset{‾}{\delta }]$ introduced in the work of Martínez-Peñas and Kschischang. The Pseudo multilinear transformations (PMT’s) are introduced and correspond to modules over $S$. These maps are strongly connected to the evaluation of polynomials in $S$. A general product formula is obtained. PMT’s help to put some structure on the set of roots of a polynomial $f(t)\in S$.