$z$-invariant function equal to a polynomial of $z$

Question

If I have the expression $f(x,y) = \sum z^ig_i(x,y)$, i.e. a $z$-invariant function equal to a polynomial of $z$ whose coefficients do not depend on $z$, and it needs to hold for all $z$, can this only be true for $f = g_i = 0$ for all $i$?