Polynomial mapping
In algebra, a polynomial map or polynomial mapping [math]\displaystyle{ P: V \to W }[/math] 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
- [math]\displaystyle{ P(v) = \sum_{i_1, \dots, i_n} \lambda_{i_1}(v) \cdots \lambda_{i_n}(v) w_{i_1, \dots, i_n} }[/math]
where the [math]\displaystyle{ \lambda_{i_j}: V \to k }[/math] are linear functionals and the [math]\displaystyle{ w_{i_1, \dots, i_n} }[/math] are vectors in W. For example, if [math]\displaystyle{ W = k^m }[/math], then a polynomial mapping can be expressed as [math]\displaystyle{ P(v) = (P_1(v), \dots, P_m(v)) }[/math] where the [math]\displaystyle{ P_i }[/math] 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
References
- Claudio Procesi (2007) Lie Groups: an approach through invariants and representation, Springer, ISBN 9780387260402.
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