$t^2 + t$ is a polynomial, then what is $t^{-2} + t^{-1}$ called? $t^2 + t$ is a polynomial.
What is
$t^{-2} + t^{-1}$ called?
More generally,
$t^{-n} + t^{-(n-1)} + \cdots + t^{-2} + t^{-1} + c$.
 A: For the sake of completeness, I summarize the comments above:
As pointer out by Torsten and manthanomen, there are specific names (Laurent polynomial) for such functions. However, these are not widely known, so I agree with Brian and Deepak and suggest using a more general concept called rational function:

A function $f(x)$ is called a rational function if and only if it can be written in the form $f(x) = \frac{P(x)}{Q(x)}$ where $P$ and $Q$ are polynomial functions of $x$ and $Q$ is not the zero function.

If you do not want to call it rational function and want to define these functions merely through the regular polynomials then saying $f(t) = P\left(\tfrac{1}{t}\right)$ as suggested by N. S. is the way to go.
Finally, as pointed out by David, $$(t+t^2)^{-1} = \frac{1}{t + t^2} \ne \frac{1 + t}{t^2} = t^{-1} + t^{-2},$$ hence not an inverse number in the usual meaning, and as noticed by symplectomorphic, $f(t) = t + t^2$ as a function does not have an inverse function in the usual meaning because it is not injective: $f(0) = f(-1) = 0$.
A: Let $k$ be a field. The linear map $\overline{\cdot}: k[t, t^{-1}] \to k[t, t^{-1}]$ which sends $t$ to $t^{-1}$ is often called the "bar involution'' in certain subjects in algebra (for example, Hecke algebras and quantum groups). So $t^{-2} + t^{-1}$ could be called "the image of $t^2 + t$ under the bar involution", i.e., $\overline{t^2 + t} = t^{-2} + t^{-1}$. Of course, as one of the commenters has noted, the image under the bar involution is not the multiplicative inverse of the polynomial.
A: In some areas of algebraic combinatorics, the related operation $f(q) \mapsto q^{\mathrm{deg}(f)} f(q^{-1})$ is called  "reversal". Here one usually has $f(0) \neq 0$, making it an involution.
