Polynomial mapping

In algebra, a polynomial map or polynomial mapping ${\displaystyle P:V\to W}$ 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

${\displaystyle P(v)=\sum _{i_{1},\dots ,i_{n}}\lambda _{i_{1}}(v)\cdots \lambda _{i_{n}}(v)w_{i_{1},\dots ,i_{n}}}$

where the ${\displaystyle \lambda _{i_{j}}:V\to k}$ are linear functionals and the ${\displaystyle w_{i_{1},\dots ,i_{n}}}$ are vectors in W. For example, if ${\displaystyle W=k^{m}}$, then a polynomial mapping can be expressed as ${\displaystyle P(v)=(P_{1}(v),\dots ,P_{m}(v))}$ where the ${\displaystyle P_{i}}$ 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.