{{Short description|Type of functions in algebra}} {{one source |date=May 2024}} In algebra, a '''polynomial map''' or '''polynomial mapping''' <math>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>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>\lambda_{i_j}: V \to k</math> are linear functionals and the <math>w_{i_1, \dots, i_n}</math> are vectors in ''W''. For example, if <math>W = k^m</math>, then a polynomial mapping can be expressed as <math>P(v) = (P_1(v), \dots, P_m(v))</math> where the <math>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 == *Polynomial functor
== References == *Claudio Procesi (2007) ''Lie Groups: an approach through invariants and representation'', Springer, {{isbn|9780387260402}}.
Category:Algebra
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