Slice regularity in several variables

We introduce a class of slice regular functions of several variables on a real alternative algebra. We generalize the definition given by Ghiloni and Perotti for functions of one variable on real alternative algebras. In the quaternionic case, several variables have been considered recently by Colombo, Sabadini and Struppa. Our approach to the definition of slice functions, which is based on the concept of {stem functions}, is the same as the one adopted by these authors. However, the condition of regularity is different, and allows to include in our class, in particular, the family of ordered polynomials in several variables. We prove some basic properties and results of slice and slice regular functions and give examples to illustrate this function theory.