  We introduce Wirtinger operators for functions of several quaternionic
variables. These operators are real linear partial differential operators which
behave well on quaternionic polynomials, with properties analogous to the ones
satisfied by the Wirtinger derivatives of several complex variables. Due to the
non-commutativity of the variables, Wirtinger operators turn out to be of
higher order, except the first ones that are of the first order. In spite of
that, these operators commute each other and satisfy a Leibniz rule for
products. Moreover, they characterize the class of slice-regular polynomials,
and more generally of slice-regular quaternionic functions. As a step towards
the definition of the Wirtinger operators, we provide Almansi-type
decompositions for slice functions and for slice-regular functions of several
variables. We also introduce some aspects of local slice analysis, based on the
definition of locally slice-regular function in any open subset of the
$n$-dimensional quaternionic space.
