About: Polynomial mapping

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In algebra, a polynomial map or polynomial mapping 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 where the are linear functionals and the are vectors in W. For example, if , then a polynomial mapping can be expressed as where the are (scalar-valued) polynomial functions on V. (The abstract definition has an advantage that the map is manifestly free of a choice of basis.)

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  • In algebra, a polynomial map or polynomial mapping 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 where the are linear functionals and the are vectors in W. For example, if , then a polynomial mapping can be expressed as where the 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. (en)
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  • In algebra, a polynomial map or polynomial mapping 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 where the are linear functionals and the are vectors in W. For example, if , then a polynomial mapping can be expressed as where the are (scalar-valued) polynomial functions on V. (The abstract definition has an advantage that the map is manifestly free of a choice of basis.) (en)
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  • Polynomial mapping (en)
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