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.

Then $$f \left({x_1, x_2, \ldots, x_m}\right)$$ and $$g \left({x_1, x_2, \ldots, x_m}\right)$$ are:


 * identically equal iff $$f \left({x_1, x_2, \ldots, x_m}\right) = g \left({x_1, x_2, \ldots, x_m}\right)$$.
 * formally equal iff the corresponding coefficients of $$f$$ and $$g$$ are equal.

Theorem
$$f$$ and $$g$$ are identically equal iff $$f$$ and $$g$$ are formally equal.

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