I know that a 2-sphere is a smooth manifold because I can define the smooth map $(x_1, x_2) \to (x_1, x_2, \sqrt{1-x_1^2-x_2^2})$ for $x_1^2 + x_2^2 < 1$ that parametrizes the $x_3 > 0$ region of $S^2$. Similarly, by interchanging the roles of $x_1, x_2, x_3$ and changing the signs of the variables, we can obtain smooth parametrizations of regions $x> 0$, $x<0$, and $z < 0$ of $S^2$. Since these smooth maps cover $S^2$, and the compositions of smooth maps are smooth, then all transition maps of $S^2$ with this atlas are smooth, so $S^2$ is a smooth manifold.
Now I am just wondering whether the 2-sphere is a rational algebraic variety over $\mathbb{R}$. i.e. is it a manifold where the transition maps are ratios of polynomials? A transition map from the $xy$ plane to the $yz$ plane with the atlas defined above would be $(x,y) \to (y, \sqrt{1-x^2-y^2})$, but this map is not a ratio of polynomials. I wonder if there exists an atlas that covers $S^2$ such that all transition maps are ratios of polynomials. My intuitive guess is no, but I don't know how exactly I can show it. Can someone help? Thanks in advance!
