This is perhaps unanswerable,
or perhaps I am too algebraically ignorant to phrase it cogently, but:

> Is there some identifiable reason that polynomials over
$\mathbb{C}$,
$\mathbb{R}$, $\mathbb{Q}$, $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$
are so pervasively useful in mathematics?

Is it because polynomials are in some sense the most natural functions
defined on a field?
I know that *every* function over a finite field $\mathbb{F}^n \to \mathbb{F}$ is a polynomial.
And, by the
[Stone–Weierstrass theorem](http://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem),
every continuous function on an interval can be approximated by a polynomial.
Is this the universal aspect of polynomials that
"explains" their ubiquity?

Even tropical polynomials, which employ
[alternative addition/multiplication operations](http://en.wikipedia.org/wiki/Tropical_geometry)
forming a semiring, are proving useful.

I'd appreciate your insights!