Consider the algebraic group $G = \textrm{GL}_n(k)$ over an algebraically closed field $k$. We may consider the underlying set of $G$ to be the closed set $V( Y\cdot \det(X_{ij})-1)$ in $k^{n^2+1}$. Now, the regular functions $G \rightarrow k$ (that is, the elements of $\mathcal O_G(G)$) consist of those functions $f$ which are regular, that is they locally be represented by rational functions. For example, $\frac{1}{\det(x_{ij})} \in \mathcal O_G(G)$. However, the big result about regular functions is that when it comes to affine varieties, the ring of regular functions $\mathcal O_G(G)$ actually consists of polynomial functions $f: G \rightarrow k$.
Now, $\frac{1}{\det(x_{ij})}$ is not in the form of the polynomial. But we know that there should exist some polynomial function $f = f(x_{ij},y): k^{n^2+1} \rightarrow k$ such that $f$ agrees with $\frac{1}{\det(x_{ij})}$ on $G = V( Y\cdot \det(X_{ij})-1)$. What function could that be? Is it obvious, or does one need to go through the proof of the result about regular functions on $\mathcal O_G(G)$ to find it?