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, ISBN 9780387260402.