In algebra, a polynomial map or polynomial mapping
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 where the
, then a polynomial mapping can be expressed as
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.
This algebra-related article is a stub.