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 $3$?
 A: In abstract algebra, we write the set of all polynomials with coefficients in a ring $R$ as $R[x]$.
Here "polynomials" means expressions of the form $$a_0+a_1x+a_2x^2+\cdots +a_nx^n$$ where $a_0,\ldots,a_n\in R$ and $n$ is finite (Note: if you don't know what a ring is, just think of the $a$'s as numbers). So, in this context, your expression $x^{-4}+x^3$ isn't in a polynomial, because all the powers of $x$ have to be non-negative.
We can generalize this construction, though.  The first thing we can do is drop the requirement that $n$ must be finite.  If we do this, we get $R[[x]]$, the set of formal power series in $R$.
A futher generalization is the set of formal Laurent series in $R$, denoted $R(\hspace{-0.5pt}(x)\hspace{-0.5pt})$, and this is a setting in which we can answer your question.  Formal Laurent series have the form 
$$\sum_{n\in \mathbb{Z}}a_nx^n$$ 
where $a_n=0$ for all but finitely many negative $n$.  In other words, formal Laurent series are formal power series which are allowed to have a finite number of negative exponents too.
The order of a formal Laurent series is defined as the smallest $n$ such that $a_n\not= 0$.  This is kind of like the degree of a polynomial, but for negative integers.  The degree of a formal Laurent series is defined in the same way as the degree of a polynomial, though the degree may not exist (since all of the $a_n$ for $n>0$ are still allowed to be nonzero).
So, considered as a formal Laurent series, we would say that $x^{-4}+x^3$ has degree $3$ and order $-4$.
A: For the sake of completeness, I would like to add that this generalization of polynomials is called a Laurent polynomial. This set is denoted $R[x,x^{-1}]$.
A: 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.
