Polynomial mapping

In algebra, a polynomial mapping $P:V\to W$ 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

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

where $L_{j}:V\to k$ are linear functionals. For example, if $W=k^{m}$, then it can also be expressed as $P(v)=(P_{1}(v),\dots ,P_{m}(v))$ where $P_{i}$ are (scalar-valued) polynomial functions on V.

One of the fundamental outstanding question regarding polynomial mapping is Jacobian conjecture, which concerns the sufficiency of a polynomial mapping to be invertible.