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!



Why are polynomials so useful in mathematics?
Polynomials are, essentially by definition, precisely the operations one can write down starting from addition and multiplication. More formally, polynomials with coefficients in a commutative ring $R$ are precisely the morphisms in the Lawvere theory of commutative $R$-algebras. So in some sense caring about polynomials is equivalent to caring about commutative rings and, more generally, commutative algebras. See, for example, [this blog post](http://qchu.wordpress.com/2013/06/09/operations-and-lawvere-theories/) for some details; in particular, that blog post makes precise the assertion that polynomials are not only the most natural but in fact the only natural operations on commutative $R$-algebras. 


