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\begin{document}

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\section*{Laurent series}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{laurent_series_as_distributions}{Laurent series as distributions}\dotfill \pageref*{laurent_series_as_distributions} \linebreak
\noindent\hyperlink{algebraic_closure}{Algebraic closure}\dotfill \pageref*{algebraic_closure} \linebreak
\noindent\hyperlink{function_field_analogy}{Function field analogy}\dotfill \pageref*{function_field_analogy} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\emph{Laurent series} generalize [[power series]] by allowing both positive and negative powers. In particular, \emph{Laurent series} with complex coefficients generalize [[Taylor series]] of analytic functions to [[meromorphic functions]]. A Laurent series for a meromorphic function $f(z)$ at finite $z\in\mathbb{C}$ has the form

\begin{displaymath}
f(z) = \sum_{n=k}^{\infty}f_n z^n,
\end{displaymath}
where $k$ is merely constrained to be finite and is often negative.

Or, in some contexts one wants to take $k = -\infty$. Here such a formal sum (with powers extending infinitely in both directions) is suggestive notation for an element belonging to the dual $\prod_{n \in \mathbb{Z}} R \cdot z^n$ of a ring $\oplus_{n \in \mathbb{Z}} R \cdot z^n$ (see Laurent polynomials below). In such contexts, Laurent series can be likened to distributions, i.e., functionals on the algebra of functions $\mathbb{Z} \to R$ with compact support.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
A \emph{Laurent series} in one variable $z$ over a commutative unital ring $k$ is a [[doubly infinite series]]

\begin{displaymath}
f(z) = \sum_{n=-\infty}^{\infty} f_n z^n ,
\end{displaymath}
where $f_n\in k$. Equivalently: a Laurent series is a function $\mathbb{Z} \to k: n \mapsto f_n$. The $k$-module of Laurent series is denoted $k[ [z, z^{-1}] ]$.

\end{defn}
A \emph{Laurent polynomial} is a Laurent series for which all but finitely many $f_n$ are zero. Laurent polynomials form a ring which may be described as $k[z, z^{-1}]$ or abstractly as $k[x, y]/(x y - 1)$. Observe that Laurent series in the generality discussed here do not analogously form a ring: the obvious definition of the coefficients of the product $h(z) = f(z)g(z)$ of two Laurent series $f(z), g(z)$,

\begin{displaymath}
h_n = \sum_{k \in \mathbb{Z}} f_k g_{n-k},
\end{displaymath}
doesn't make sense in general (although it could sometimes make sense in topological contexts where some such infinite sums can converge, as in the case $k = \mathbb{C}$).

\begin{remark}
\label{}\hypertarget{}{}
The $k$-vector space of Laurent series does however form a module over the ring of Laurent polynomials, i.e., if $f(z) \in k[z, z^{-1}]$ and $g(z)$ is a Laurent series, then the product $h(z) = f(z)g(z)$ as defined above always makes sense.

\end{remark}
In general, questions of convergence are treated as separate issues. In complex analysis, the Laurent series $\sum_{n \in \mathbb{Z}} a_n z^n$ describes a meromorphic function in a neighborhood around the point $z = 0$ (possibly with a pole there) if all but finitely many negatively indexed terms are zero. Similarly, series of the form $\sum_{n = -N}^{\infty} a_n (z-a)^n$ describe meromorphic functions in a neighborhood of $z=a$ with poles of order at most $N$.

On the other hand, in algebra one often hears of the ring of \emph{formal Laurent series}. Here, the presence of the word ``ring'' signifies that we are restricting the coefficients so that multiplication makes sense. Thus,

\begin{defn}
\label{ring}\hypertarget{ring}{}
The \emph{ring of formal Laurent series} over a [[commutative ring]] $A$ in an indeterminate $x$ consists of Laurent series $\sum_{n \in \mathbb{Z}} f_n z^n$, with $f_n \in A$ but where all but finitely many $f_n$ for $n \lt 0$ vanish.

\end{defn}
Multiplication defined as above clearly makes sense. If $A$ is a field $k$, then this ring is usually denoted $k((x))$ and is in fact a [[field]]; indeed it is the [[field of fractions]] of the ring $k[ [x] ]$ of [[formal power series]], where $k[ [x] ]$ is often viewed as a discrete [[valuation ring]].

\begin{remark}
\label{restrict}\hypertarget{restrict}{}
It would perhaps be clearer if we used the term ``restricted Laurent series'' to cover the Laurent series considered in Definition \ref{ring}, and let ``Laurent series'' be the term that covers doubly infinite series.

\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{laurent_series_as_distributions}{}\subsubsection*{{Laurent series as distributions}}\label{laurent_series_as_distributions}

Another point of view on Laurent series is given in the following alternative definition.

\begin{defn}
\label{}\hypertarget{}{}
A Laurent series over $k$ is a $k$-linear functional

\begin{displaymath}
\phi: k[z, z^{-1}] \to k
\end{displaymath}
on the algebra of Laurent polynomials.

\end{defn}
The associated series is $\sum_{n \in \mathbb{Z}} \phi(z^n)z^n$. But here we emphasize the point of view that Laurent series have a distribution-like character, with Laurent polynomials being considered a space of functions $\mathbb{Z} \to k$ with compact (finite) support, via the evident inclusion

\begin{displaymath}
k[z, z^{-1}] \cong \oplus_{n \in \mathbb{Z}} k \cdot z^n \hookrightarrow \prod_{n \in \mathbb{Z}} k \cdot z^n = k^{\mathbb{Z}}.
\end{displaymath}
Of particular interest as a Laurent series is the formal Dirac distribution,

\begin{displaymath}
\delta(z) = \sum_{n \in \mathbb{Z}} z^n,
\end{displaymath}
which intuitively is like the Fourier transform $\sum_{n = -\infty}^\infty e^{i n x}$ of the pointwise multiplicative identity $\mathbf{1}$ given by $\mathbf{1}(n) = 1$ for all $n$. It has the following property:

\begin{itemize}%
\item For $f(z)$ a Laurent polynomial, $f(z)\delta(z) = f(1)\delta(z)$.

\end{itemize}
Indeed, notice that $\delta(z)$ is the distribution $p \mapsto p(1)$ which evaluates a Laurent polynomial at the multiplicative identity $z=1$.

More generally, for $a \in k^\ast$ an invertible unit, the series $\delta(a z) = \sum_{n \in \mathbb{Z}} a^n z^z$ satisfies

\begin{itemize}%
\item $f(z)\delta(a z) = f(a^{-1})\delta(a z)$.

\end{itemize}
Just as it doesn't make sense in general to multiply distributions (at least not without heavy qualifications, as in the theory of \hyperlink{Col}{Colombeau}), we cannot make sense of expressions like $\delta(z)^2$. However, one can meaningfully work with derivatives of distributions, so we have for example

\begin{displaymath}
\delta'(z) = \sum_{n \in \mathbb{Z}} n z^n
\end{displaymath}
which has the property

\begin{itemize}%
\item $f(z)\delta'(z) = f(1)\delta'(z) - f'(1)\delta(z)$.

\end{itemize}
Indeed, there is a formal calculus of the Dirac distribution that plays an important role in the theory of vertex operator algebras. See for example the treatment in \hyperlink{FLM}{Frenkel, Lepowsky, Meurman}.

One can also work with \emph{convolution products} $h(z) = (f \ast g)(z)$, defined by multiplying coefficients degree-wise:

\begin{displaymath}
h(z) = \sum_{n \in \mathbb{Z}} f_n g_n z^n.
\end{displaymath}
\hypertarget{algebraic_closure}{}\subsubsection*{{Algebraic closure}}\label{algebraic_closure}

\begin{theorem}
\label{Puiseux}\hypertarget{Puiseux}{}
If $K$ is [[algebraically closed field|algebraically closed]] and has [[characteristic]] 0, then the [[algebraic closure]] of the field of (restricted) Laurent series $K((x))$ over $K$ is the field of [[Puiseux series]] over $K$.

\end{theorem}
Here ``restricted'' refers to Remark \ref{restrict}. See \emph{[[Puiseux series]]} for more details on this result.

\hypertarget{function_field_analogy}{}\subsubsection*{{Function field analogy}}\label{function_field_analogy}

[[!include function field analogy -- table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Igor Frenkel, James Lepowsky, and Arne Meurman, \emph{Vertex Operator Algebras and the Monster}, Volume 134 in Pure and Applied Mathematics, Academic Press 1988.

\end{itemize}
\begin{itemize}%
\item \href{http://math.rutgers.edu/~sdurst/DILS.html}{Doubly Infinite Laurent Series}, lectures for formal Laurent series in [[vertex operator algebra]] context
\item Alexander Zheglov, \emph{Wild division algebras over Laurent series fields}, \href{http://arxiv.org/abs/math/0503637}{math.NT/0503637}

\end{itemize}
For discussion of products of distributions, see

\begin{itemize}%
\item J.F. Colombeau, \emph{Multiplication of distributions}, Bull. Amer. Math. Soc. (N.S.) Volume 23, Number 2 (1990), 251-268. (\href{http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183555881}{web})

\end{itemize}


\end{document}