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Polynomial mapping

Type of functions in algebra

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated July 21, 2026

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.

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This article is adapted from the Wikipedia article “Polynomial mapping”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Polynomial functor

In algebra, a polynomial functor is an endofunctor on the category V {\displaystyle {\mathcal {V}}} of finite-dimensional vector spaces that depends polynomially on vector spaces. For example, the symmetric powers V ↦ Sym n ⁡ {\displaystyle V\mapsto \operatorname {Sym} ^{n}(V)} and the exterior powers V ↦ ∧ n ( V ) {\displaystyle V\mapsto \wedge ^{n}(V)} are polynomial functors from V {\displaystyle {\mathcal {V}}} to V {\displaystyle {\mathcal {V}}} ; these two are also Schur functors.