Polynomial mapping

In algebra, a polynomial map or polynomial mapping formula_1 between vector spaces over an infinite field "k" is a polynomial in linear functionals with coefficients in "W"; i.e., it can be written as
where the formula_3 are linear functionals and the formula_4 are vectors in "W". For example, if formula_5, then a polynomial mapping can be expressed as formula_6 where the formula_7 are (scalar-valued) polynomial functions on "V". (The abstract definition has an advantage that the map is manifestly free of a choice of basis.)
When "V", "W" are finite-dimensional vector spaces and are viewed as algebraic varieties, then a polynomial mapping is precisely a morphism of algebraic varieties.
One fundamental outstanding question regarding polynomial mappings is the Jacobian conjecture, which concerns the sufficiency of a polynomial mapping to be invertible.