Suppose there is a function of $n$ variables, $f(x_1, ..., x_n)$, and we write it in terms of $n$ different variables, $g(y_1, ..., y_n)$. Would the functions have the same number of roots (over any field)?
Thanks for your help!
Suppose there is a function of $n$ variables, $f(x_1, ..., x_n)$, and we write it in terms of $n$ different variables, $g(y_1, ..., y_n)$. Would the functions have the same number of roots (over any field)?
Thanks for your help!
As requested:
Not unless the change of variables is one-to-one; for example, we can take the 1-variable function $f(x)=x^2-4$, and rewrite it in terms of the 1-variable function $g(y)=y-4$ via the change-of-variable $y=x^2$ (which is not one-to-one, since distinct values of $x$ lead to the same value of $y$). Then $f(x)=0$ has two solutions (in the real or complex numbers, say), but $g(y)=0$ has only one solution.