Polynomial mapping explained

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 asP(v) = \sum _{i _{1,} ... , i _{n}} λ _{i _{1}} (v) … λ _{i _{n}} (v) w _{i _{1,} ... , i _{n}} where the λ _{i _{j}} : V \to k are linear functionals and the w _{i _{1,} ... , 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),} ... , 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, .