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Polynomial mapping

Type of functions in algebra

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Inalgebra, a polynomial maporpolynomial mapping ${\displaystyle P:V\to W}$ between vector spaces over an infinite field k is a polynomialinlinear functionals with coefficients in k; i.e., it can be written as

${\displaystyle P($v)=\sum _{i_{1},\dots ,i_{n}}\lambda _{i_{1}}(v)\cdots \lambda _{i_{n}}(v)w_{i_{1},\dots ,i_{n}}}

where the ${\displaystyle \lambda _{i_{j}}:V\to k}$ are linear functionals and the ${\displaystyle w_{i_{1},\dots ,i_{n}}}$ are vectors in W. For example, if ${\displaystyle W=k^{m}}$, then a polynomial mapping can be expressed as ${\displaystyle P($v)=(P_{1}(v),\dots ,P_{m}(v))} where the ${\displaystyle P_{i}}$ are (scalar-valued) polynomial functionsonV. (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[edit]

• Polynomial functor

References[edit]

• Claudio Procesi (2007) Lie Groups: an approach through invariants and representation, Springer, ISBN 9780387260402.

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