Proof of the Computability of Polynomials In studying properties of polynomial functions I have read that they are computable. The usage of the word read implies that I cannot prove this statement, and withhold using learned for this reason. Additionally, I chose to ask this question here, as opposed to the cs exchange, since I am interested in the mathematicians' vantage point for this question. Without further ado I would like to know how the polynomial functions can be shown to be computable. Moreover, I am also interested in how this property of polynomial functions helps the applied and computational mathematician in their endeavors. In order that this question be deemed addressable it might be best to restrict the latter question to "favorite" usages of the polynomial functions being computable in applied and computational mathematics.

Actually, they're not.

Based on the Wikipedia article you linked, it looks like the polynomials in question are functions from $\mathbb{R}$ to $\mathbb{R}$. Such a function $f$ is computable if there is an oracle Turing machine that, when given $x \in \mathbb{R}$ as an oracle and $n$ as an input, produces the $n$-th digit of $f(x)$. Caveat: this definition is only essentially correct.

So if a polynomial has a non-computable real number as a coefficient, then it might not be a computable function. For example let $K$ be the real number corresponding to the halting set, so the $n$-th binary digit of $K$ is $1$ iff the $n$-th machine halts. Then the constant function $f(x)=K$ is not computable but is certainly a polynomial.

But if the polynomial has, for example, rational coefficients, then it is computable.