polynomial mapping

{{Short description|Type of functions in algebra}}

{{one source |date=May 2024}}

In algebra, a polynomial map or polynomial mapping $P:\; V\; \backslash to\; W$ 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

:$P(v)\; =\; \backslash sum\_\{i\_1,\; \backslash dots,\; i\_n\}\; \backslash lambda\_\{i\_1\}(v)\; \backslash cdots\; \backslash lambda\_\{i\_n\}(v)\; w\_\{i\_1,\; \backslash dots,\; i\_n\}$

where the $\backslash lambda\_\{i\_j\}:\; V\; \backslash to\; k$ are linear functionals and the $w\_\{i\_1,\; \backslash dots,\; i\_n\}$ are vectors in W. For example, if $W\; =\; k^m$, then a polynomial mapping can be expressed as $P(v)\; =\; (P\_1(v),\; \backslash dots,\; P\_m(v))$ 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.

One fundamental outstanding question regarding polynomial mappings is the Jacobian conjecture, which concerns the sufficiency of a polynomial mapping to be invertible.