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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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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. 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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