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