Let $f(x,y,z)$ be a polynomial of three variables. Then
$f=0$ (which is a two dimentional object) is smooth if and only if the system $f_x=f_y=f_z=0$ has only zero solutions.
I'm not sure if the above statment is the definition of smoothness or a theorem.
My question is if we have two polynomials F and G of four variables. Then $F=0, G=0$ together also defines a two dimentional object. How to tell if this surface is smooth or not?
${\bf F}: (x_1,\ldots,x_4) \to (F(x_1, \ldots, x_4), G(x_1, \ldots,x_4))$ is a function from $\mathbb R^4$ to $\mathbb R^2$. If ${\bf F}({\bf a}) = (0,0)$ and the Jacobian matrix $$ J = \pmatrix{ \partial F/\partial x_1 & \ldots \partial F/\partial x_4\cr \partial G/\partial x_1 & \ldots \partial G/\partial x_4\cr} $$ has rank $2$ at $\bf a$, then we can take two of the four variables such that the submatrix corresponding to their columns has rank $2$. Let's say these are $x_1$ and $x_2$. The Implicit Function Theorem then says there are smooth functions $u_3(x_1, x_2)$ and $u_4(x_1,x_2)$ with $u_3(a_1, a_2) = a_3$ and $u_4(a_1, a_2) = a_4$ such that in a neighbourhood of $\bf a$, the surface ${\bf F}({\bf x}) = 0$ is given parametrically by $(x_1, x_2, u_3(x_1, x_2), u_4(x_1, x_2))$ in a neighbourhood of $(x_1, x_2) = (0,0)$, and in particular is smooth.