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\begin{document}

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\section*{Novikov field}

\hypertarget{novikov_fields}{}\section*{{Novikov fields}}\label{novikov_fields}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{characterizations}{Characterizations}\dotfill \pageref*{characterizations} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{Novikov field} (or Novikov ring) is a field of generalized [[formal power series]] that allows non-integer exponents. In contrast to a [[Hahn series]] field, in a Novikov field the series are only $\omega$-long rather than transfinitely long. This requires a ``left-finiteness'' condition on the exponents to ensure closure under multiplication.

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

There are many variant definitions of Novikov fields in the literature, see e.g. \href{https://mathoverflow.net/q/13203}{this question}. Here we give a reasonable-seeming general and abstract definition.

Let $k$ be a [[commutative ring]] and $G$ a [[linear order|linearly]] [[ordered abelian group]].

\begin{udefn}
The \textbf{Novikov ring} of $k$ with value group $G$ is the ring of functions $f:G\to k$ such that for any $y\in G$ the set $\{ x\in G \mid x \lt y \wedge f(x)\neq 0\}$ is finite. Such functions are added pointwise, and multiplied by the formula

\begin{displaymath}
(f\cdot g)(z) = \sum_{x+y=z} f(x) \cdot g(y).
\end{displaymath}
\end{udefn}
Left-finiteness of the support of $f$ and $g$ implies that the above sum is finite. Specifically, if $g\neq 0$ then since $G$ is totally ordered, there is a least $y_0$ such that $g(y_0)\neq 0$. Then the set $\{ x\mid x \le z-y_0 \wedge f(x)\neq 0\}$ is finite, and hence so is its subset $\{ x \mid \exists y. x+y = z \wedge f(x)\neq 0 \wedge g(y)\neq 0 \}$. Note that this depends on the fact that $G$ is totally ordered and a group; a partially ordered monoid would not suffice.

Notationally, we write such a function as $\sum_{x\in G} f(x)\, t^x$ for $t$ a formal variable. If $k$ is a [[field]], then so is the Novikov ring.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item When $G=\mathbb{R}$, the Novikov field is sometimes called the ``universal Novikov field''.


\item When $G=\mathbb{Q}$ and $k =\mathbb{R}$, the Novikov field is known as the \textbf{Levi-Civita field}.



\end{itemize}
\hypertarget{characterizations}{}\subsection*{{Characterizations}}\label{characterizations}

The Novikov field embeds into the [[Hahn series]] field $k[[t^G]]$. It can (probably) be characterized therein as

\begin{itemize}%
\item The set of Hahn series with order type $\omega$ that converge to themselves in the valuation topology.


\item The closure of the field $k(t^G)$ of [[generalized rational functions]] inside the Hahn series field.



\end{itemize}
It can also (probably) be characterized abstractly as

\begin{itemize}%
\item The Cauchy completion of $k(t^G)$ in its valuation [[uniform space|uniformity]].


\item The completion of $k(t^G)$ as a valued field, i.e. the unique (up to isomorphism) dense valued field extension without proper dense valued field extension.



\end{itemize}
\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\begin{itemize}%
\item The universal Novikov field of $\mathbb{R}$ is a natural context in which to relate [[magnitude homology]] of finite [[metric spaces]] to their [[magnitude]]. (Hahn series also suffice, but all the action actually takes place in the Novikov field.)


\item In the [[Fukaya category]], the chain complexes defining Hom's between objects are defined over a Novikov ring.



\end{itemize}
\hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages}

Other rings of generalized power series include:

\begin{itemize}%
\item [[Puiseux series]]
\item [[Hahn series]]
\item [[Ribenboim power series]]

\end{itemize}
Hahn series are a special kind of Ribenboim power series, but Puiseux and Novikov series are not. However, they are all instances of the linearization of a [[finiteness space]].

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

\begin{itemize}%
\item \href{https://en.wikipedia.org/wiki/Novikov_ring}{Wikipedia: Novikov ring}
\item \href{https://en.wikipedia.org/wiki/Quantum_cohomology}{Wikipedia: quantum cohomology}
\item \href{https://mathoverflow.net/q/13203}{MathOverflow: definitions of Novikov field}

\end{itemize}
[[!redirects Novikov field]] [[!redirects Novikov fields]] [[!redirects Novikov ring]] [[!redirects Novikov rings]] [[!redirects Novikov series]] [[!redirects universal Novikov field]] [[!redirects universal Novikov fields]] [[!redirects universal Novikov ring]] [[!redirects universal Novikov rings]] [[!redirects Levi-Civita field]]



\end{document}