Equality of Polynomials

Definition
Let $\left({k, +, \circ}\right)$ be an infinite field.

Let $k \left[{\left\{ {X_j: j \in J}\right\} }\right]$ be the ring of polynomial forms in the indeterminates $\left\{ {X_j: j \in J}\right\}$.

Let $f, g \in k \left[{\left\{ {X_j: j \in J}\right\} }\right]$

Then $f$ and $g$ are:


 * equal as functions if the polynomial functions associated to $f$ and $g$ are equal as functions, that is:


 * $\forall x \in k^J: f \left({x}\right) = g \left({x}\right)$

where $k^J$ is the free module on $J$.


 * equal as forms if the functions $M \to k$ from the free commutative monoid to $k$ which define $f$ and $g$ are equal as functions.

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

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