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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 $3$?

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A polynomial only contains positive, integral indices. So $x^{-4}+x^{3}$ is not a polynomial. –  Shaktal Nov 6 '12 at 13:25
A polynomial has only positive exponents. The degree of a rational function, that is a quotient of two polynomials, in your case $(x^7 + 1)/x^4$ is usually defined as the difference of the degrees of the involved polynomials. –  martini Nov 6 '12 at 13:26
@Shaktal : Thanks for adding the proper editing tags. –  bonCodigo Nov 6 '12 at 13:43
@martini : Thank you for guiding to the correct definition. So the expression I have is a rational function. If the resulting expression of the difference of a rational function is polynomial then degree can be found using polynomial exponential rule. In my case, the difference is resulting in a rational function. Can I say the degree is 3 in that case? Because upper degree of numerator is 7, upper degree of denominator is 4, the difference is 3... –  bonCodigo Nov 6 '12 at 14:19
@bonCodigo Yes. –  martini Nov 6 '12 at 14:20

2 Answers 2

up vote 3 down vote accepted

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 aring is, just think of the a's as numbers.) So, your expression $x^{-4}+x^3$ isn't in $R[x]$.

We can generalize this notion, though. The first thing we can do is drop the requirement that $n$ must be finite. If we do this, we obtain $R[[x]]$, the set of formal power series in $R$.

A futher generalization is the set of formal Laurent series in $R$, denoted $R((x))$, and this is what we need to talk about your question. Formal Laurent series have the form $$\sum_{n\in \mathbb{Z}}a_nx^n$$ where $a_n=0$ all but finitely many negative $n$. In other words, formal Laurent series are formal power series which are allowed to have finitely many 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 the same 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 $-4$.

Remark. I should mention that $R[x]$, $R[[x]]$, and $R((x))$ are actually more than just sets. They are rings, which just means that you can add and multiply their elements.

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For the sake of completeness, I would like to add that this generalization of polynomials is called a Laurent polynomial. This set is denoted $R[x,x^{-1}]$.

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