Slice regular functions in several variables

Abstract In this paper, we lay the foundations of theory slice regular functions in several (non-commuting) variables ranging any real alternative $$^*$$ ? -algebra, including quaternions, octonions and Clifford algebras. This higher dimensional function is an extension classical holomorphic complex variables. It based on construction a family commuting structures $${\mathbb {R}}^{2^n}$$ R 2 n . One relevant aspects validity Cauchy-type integral formula existence ordered power series expansions. The includes all polynomials with right coefficients algebra. We study dimension zero set quaternionic octonionic cases give some results about coefficients. particular, show that nonconstant polynomial always has non empty set.