The degree of a polynomial which also has negative exponents. In theory, we define the degree of a polynomial as the highest exponent it holds. 
However when there are negative and positive exponents are present in the function, I want to know the basis that we define the degree.  Is the order of a polynomial degree expression defined by the highest magnitude of available exponents?
For example in $x^{-4} + x^{3}$, is the degree $4$ or $3$?

Your question is essentially: how does one define the degree of a Laurent polynomial?
My guess is that the best of of doing this is to move from degrees to "degree pairs." Let me explain this for polynomials first. 
Given a polynomial $$P(x) = \sum_{n:\mathbb{N}} a_n x^n,$$ define the degree pair of $P$ to be the pair $(i,j)$ where $i$ is the least natural number such that $a_i$ is non-zero, and $a_j$ is the greatest natural number such that $a_j$ is non-zero. Denote this $\mathrm{dpa}(P)$. For example:
$$\mathrm{dpa}(-3x^5+x^2+x) = (1,5)$$
This extends straightforwardly to Laurent polynomials, and seems to be relatively well-behaved.
