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) element 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 element 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 δ_i :K^n->K, defined by δ_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.

invertible polynomial map | Jacobian | Jacobian conjecture | map | polynomial