Equality of Polynomials

Definition
Let $$f \left({X_1, X_2, \ldots, X_m}\right)$$ and $$g \left({X_1, X_2, \ldots, X_m}\right)$$ be polynomials in $$m$$ variables over a field $$K\subseteq \C$$ of characteristic zero.

Then $$f \left({X_1, X_2, \ldots, X_m}\right)$$ and $$g \left({X_1, X_2, \ldots, X_m}\right)$$ are:


 * equal as functions iff $$f \left({x_1, x_2, \ldots, x_m}\right) = g \left({x_1, x_2, \ldots, x_m}\right)$$ for all $$x_1,\ldots,x_n\in K$$.
 * equal as polynomials iff the corresponding coefficients of $$f$$ and $$g$$ are equal.

Theorem
$$f$$ and $$g$$ are equal as polynomials iff $$f$$ and $$g$$ are equal as functions.

Thus we can say $$f = g$$ without ambiguity as to what it means.