ag.algebraic geometry - A rational point in the scheme of pointed degree n rational functions [0912.2227]

The following question is related to "Remark 2.2" in Christophe Cazanave's paper "Algebraic homotopy classes of algebraic functions". I decided to add the arxiv article-id to the questions title to invite other people who like to study this article to do the same. My hope is that this will lead to a culture of discussing arxiv articles on the overflow.

Question:
Let $F_n$ be the open subscheme of $mathbb{A}^{2n}=mathrm{Spec}(k[a_{0},ldots,a_{n-1},b_{0},ldots,b_{n-1}])$ complementary to the hypersurface of equation $res_{n,n}(X^{n}+a_{n-1}X^{n-1}+ldots+a_{0},b_{n-1}X^{n-1}+ldots+b_{0})$. Let $R$ be a ring. The claim is that an $R$-point of $F_{n}$ is a pair $(A,B)$ of polynomials of $R[X]$, where $A$ is monic of degree $n$, $B$ is of degree strictly less than $n$ and the scalar $res_{n,n}(A,B)$ is invertible. How can I see that a morphism $mathrm{Spec}(R)rightarrow F_n$ gives (and is the same as) a pair of polynomials in $R[X]$?