Equality of Polynomials
From ProofWiki
Definition
Let $(k, +, \circ)$ be an infinite field.
Let $k\left[\{ X_j : j \in J \}\right]$ be the ring of polynomial forms in the indeterminates $\{ X_j : j \in J \}$.
Let $f,g\in k\left[\{ X_j : j \in J \}\right]$
Then $f$ and $g$ are:
- equal as functions if the polynomial functions associated to $f$ and $g$ are equal as functions, that is:
- $\forall x\in k^J,\ f(x)=g(x)$
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 if and only if $f$ and $g$ are equal as functions.
Thus we can say $f = g$ without ambiguity as to what it means.
Proof
Proof missing. Also, I am not sure how general this result can be made. My suspicion is that if a comm. ring with $1$, $R$ has no idempotents save $0$ and $1$, then the result continue to hold, but not sure at the moment.
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