# Equality of Polynomials

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This page has been identified as a candidate for refactoring of medium complexity.In particular: Expand for full definitions of polynomials. Separate out definition from proof. Replace the original definition of polynomials from before it became obscured with the Rong of Polynomial Forms notation.Until this has been finished, please leave
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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. |