Definition 9.5 (Polynomial map). Let $k$ be a field. We say a function $\varphi :K^m\to K^m$ is a polynomial map if

$$\mathrm{\Phi }(\overline{x})=(p_1(x_1,\dots ,x_m),\dots ,p_n(x_1,\dots ,x_m))$$

where each $p_1,\dots ,p_n$ is a polynomial.