Equality of Polynomials

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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 if and only if $f$ and $g$ are equal as functions.

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

This article, or a section of it, needs explaining. In particular: In the exposition, the term was "equal as forms", but it has now morphed into "equal as polynomials". Needs to be resolved. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Proof

This theorem requires a proof. In particular: 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. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.