= Polynomial function
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A polynomial function is the <function> obtained by evaluating a <polynomial> on a specified <ring> or <field>. Distinct <polynomials> can induce the same polynomial function over a <finite field>; for example, $x^p$ and $x$ agree on $\mathbb F_p$.
Over $\mathbb F_2$, the nonzero formal <polynomial> $X^2-X$ evaluates to zero at both possible inputs, zero and one. It therefore induces the same <polynomial function> as the zero <polynomial>, despite having different <coefficients> and <polynomial degree>. Over an infinite <field>, equality at every input forces equality of <polynomials> by the <root bound for a polynomial>. The distinction matters whenever a representation is supposed to identify the function uniquely.
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