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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!

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    By showing that every combination of the original variables maps to exactly one combination of the new variables, and vice versa. Equivalently, it is enough to show that, given any values of the new variables, you can unambiguously recover the original variables (and, again, vice versa). The substitution $y = x^2$ doesn't satisfy that: for each y > 0 there are two possible values of $x$, and for y < 0 there are none.2011-07-19

1 Answers 1

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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.