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Polynomial Map

A map defined by one or more polynomials. Given a field K, a polynomial map is a map f:K^n->K^m such that for all points (x_1,...,x_n) in K^n,

f(x_1,...,x_n)=(g_1(x_1,...,x_n),...,g_m(x_1,...,x_n)),

for suitable polynomials g_1,...,g_m in K[X_1,...,X_n]. The zero set of f is the set of all solutions of the simultaneous equations g_1=...=g_m=0, and is an algebraic variety in K^n.

An example of polynomial map is the ith coordinate map delta_i:K^n->K, defined by delta_i(x_1,...,x_n)=x_i for all i=1,...,n. In the language of set theory, it is the projection of the Cartesian product K^n onto the ith factor.

Polynomial maps can be defined on any nonempty subset S of K^n. If S is an affine variety, then the set of all polynomial maps from S to K is the coordinate ring K[S] of S. If T is an affine variety of K^m, then every polynomial map f:S->T induces a ring homomorphism F:K[T]->K[S], defined by F(phi)=phi degreesf. Conversely, every ring homomorphism G:K[T]->K[S] determines a polynomial map g:S->T, where g=(G(delta_1),...,G(delta_m)).

A polynomial map f:R->R is a real-valued polynomial function. Its graph is the plane algebraic curve with Cartesian equation y=f(x).

See also

Jacobian, Invertible Polynomial Map, Jacobian Conjecture, Map, Polynomial

This entry contributed by Margherita Barile

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References

Becker, T. and Weispfenning, V. Gröbner Bases: A Computational Approach to Commutative Algebra. New York: Springer-Verlag, p. 330, 1993.

Referenced on Wolfram|Alpha

Polynomial Map

Cite this as:

Barile, Margherita. "Polynomial Map." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/PolynomialMap.html

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