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.
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.
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
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.
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.
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.
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.
Last revised on July 22, 2019 at 23:01:24. See the history of this page for a list of all contributions to it.