Polynomial mapping

In algebra, a polynomial map or polynomial mapping $$P: V \to W$$ between vector spaces over an infinite field k is a polynomial in linear functionals with coefficients in k; i.e., it can be written as
 * $$P(v) = \sum_{i_1, \dots, i_n} \lambda_{i_1}(v) \cdots \lambda_{i_n}(v) w_{i_1, \dots, i_n}$$

where the $$\lambda_{i_j}: V \to k$$ are linear functionals and the $$w_{i_1, \dots, i_n}$$ are vectors in W. For example, if $$W = k^m$$, then a polynomial mapping can be expressed as $$P(v) = (P_1(v), \dots, P_m(v))$$ where the $$P_i$$ 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.