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.

See also

• Polynomial functor

References

• Claudio Procesi (2007) Lie Groups: an approach through invariants and representation, Springer, .