Polynomial Map

A map defined by one or more polynomials. Given a field , a polynomial map is a map  such that for all points ,

for suitable polynomials . The zero set of  is the set of all solutions of the simultaneous equations , and is an algebraic variety in .

An example of polynomial map is the th coordinate map , defined by  for all . In the language of set theory, it is the projection of the Cartesian product  onto the th factor.

Polynomial maps can be defined on any nonempty subset  of . If  is an affine variety, then the set of all polynomial maps from  to  is the coordinate ring  of . If  is an affine variety of , then every polynomial map  induces a ring homomorphism , defined by . Conversely, every ring homomorphism  determines a polynomial map , where .

A polynomial map  is a real-valued polynomial function. Its graph is the plane algebraic curve with Cartesian equation .

REFERENCES:

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