Given a map between $\mathbb P^n_{\mathbb C}$ and itself, given by $$ [x_0:\dots:x_n] \mapsto [p_0(x):\dots:p_n(x)]$$ where $p_i$ are homogeneous polynomials of the same degree, how do I find the corresponding map between the Projective schemes?
I think I have to find a map between the polinomial rings, such that the counterimage of the maximal ideal $$ (\{a_ix_j-a_jx_i\}_{i,j})$$ is the maximal ideal $$ (\{p_i(a)y_j-p_j(a)y_i\}_{i,j}) $$ for every $a\in \mathbb{C}^{n+1}$, but I don't see an easy way to do so.
For example, I tried with the map between $\mathbb P^1$ $$ [x_0,x_1]\mapsto [x_0(x_0-x_1)^2:x_1^3] $$ but I got nothing. Any help?
The map
\begin{align} \mathbb{P}^n_\mathbb{C} &\to \mathbb{P}^m_\mathbb{C} \\ [x_0 : \cdots : x_n] & \mapsto [p_0(\overline{x}):\cdots:p_m(\overline{x})] \end{align}
corresponds to the ring homomorphism
\begin{align} \mathbb{C}[y_0, \ldots, y_m] &\to \mathbb{C}[x_0, \ldots, x_n] \\ y_i &\mapsto p_i(x_0, \ldots, x_m) \end{align}