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 $3$?
Best Answer
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 $-4$.