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Polynomial Map

A polynomial map is 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 set in K^n. It need not be an algebraic variety in the irreducible convention. For example, the zero set of xy is the union of the two coordinate axes.

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. For affine varieties S subset= K^n and T subset= K^m over an algebraically closed field K, the coordinate ring K[S] consists of the polynomial functions on S. Every polynomial map f:S->T induces a ring homomorphism F:K[T]->K[S] defined by F(phi)=phi degreesf that fixes the constant functions. Conversely, every such coefficient-fixing ring homomorphism G:K[T]->K[S], called a K-algebra homomorphism, determines a polynomial map g:S->T by g=(G(delta_1),...,G(delta_m)), where the delta_i are the coordinate functions on T (Milne 2024, pp. 67-68). The condition on constants is essential. For S=T={i} over K=C, their coordinate rings are both C, but complex conjugation is a ring homomorphism sending the coordinate class i to -i, which is not a point of T.

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

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

Portions of 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.Milne, J. S. Algebraic Geometry. Version 6.10, Nov. 11, 2024. https://www.jmilne.org/math/CourseNotes/AG.pdf.

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Polynomial Map

Cite this as:

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

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