Novikov fields

# Novikov fields

## Idea

A 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.

## Definition

There are many variant definitions of Novikov fields in the literature, see e.g. this question. Here we give a reasonable-seeming general and abstract definition.

Let $k$ be a commutative ring and $G$ a linearly ordered abelian group.

###### Definition

The 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

$(f\cdot g)(z) = \sum_{x+y=z} f(x) \cdot g(y).$

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.

## Examples

• When $G=\mathbb{R}$, the Novikov field is sometimes called the “universal Novikov field”.

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

## Characterizations

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

• The set of Hahn series with order type $\omega$ that converge to themselves in the valuation topology.

• The closure of the field $k(t^G)$ of generalized rational functions? inside the Hahn series field.

It can also (probably) be characterized abstractly as

• The Cauchy completion of $k(t^G)$ in its valuation uniformity.

• 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.

## Applications

• 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.)

• In the Fukaya category, the chain complexes defining Hom’s between objects are defined over a Novikov ring.

Other rings of generalized power series include:

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.