# Rational function as the restriction of a polynomial map

I'm having trouble with this homework problem.

Let $V = \mathbb{V}(f_1, \ldots, f_r) \subseteq \mathbb{A}^n, f\in\mathbb{C}[V]$. Show that $D_V(f) \rightarrow \mathbb{V}(f_1, \ldots, f_r, fx_{n+1}-1)$ given by $$(x_1,\ldots,x_n)\mapsto(x_1,\ldots,x_n,\frac{1}{f(x_1,\ldots,x_n)})$$ is a morphism by expicitly writing it as the restriction of a polynomial map $\mathbb{P}^n\rightarrow\mathbb{P}^{n+1}$.

I know I have to use the properties of projective space somehow, but I'm not sure how. I want to say something like $$[x_0:\cdots:x_n]\mapsto[x_0f^h(x_1,\ldots,x_n):\cdots:x_nf^h(x_1,\ldots,x_n):x_0^{deg(f)+1}]$$

so that I can divide through by $f$, and then when we restrict to $\mathbb{A}^n$ by setting $x_0=1$ it becomes the desired map; but I don't think that's right. I'd appreciate a push in the right direction.

(I hope this notation is standard - if not, just ask).

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You or your teacher have mixed up affine and projective space in your question. Please correct it. – Georges Elencwajg Feb 23 '12 at 7:36