# A ring of generalized power series

Let $$\Bbbk$$ be a field; I am interested in the following ring (which I suspect is a field). Its elements are formal expressions that look like

$$\sum_{n=0}^{\infty} a_n x^{b_n}$$

where $$a_n\in \Bbbk$$ and $$b_n\in \mathbb{R}$$, with $$b_n$$ strictly increasing, and $$\lim_{n\to\infty} b_n = \infty$$. (Technically, we should quotient these by some relation allowing us to insert and remove terms where $$a_n=0$$. Or we could require all the $$a_n$$'s to be nonzero, but then we'd have to include finite sums as well. Or we could represent them by functions $$a : \mathbb{R} \to \Bbbk$$ assigning a coefficient to each exponent, whose support is left-finite.) We can add and multiply these expressions in fairly evident ways; the condition on $$b_n$$ ensures that multiplication works (i.e. the resulting set of exponents can again be enumerated with order type $$\omega$$ and limit $$\infty$$).

This ring of power-series-like-objects is closely related to some others. Specifically, if $$\Bbbk=\mathbb{R}$$ then it contains the Levi-Civita field as the elements for which each $$b_n\in\mathbb{Q}$$, while it is contained in the Hahn series field $$\Bbbk[[x^{\mathbb{R}}]]$$. Note that the set of all Hahn series with order type $$\omega$$ is not closed under multiplication; this is a natural subset thereof that is. I believe that it is also the set of Hahn series with order type $$\omega$$ that converge to themselves in the valuation topology of the Hahn series field, and also that it is the closure of the field $$\Bbbk(x^{\mathbb{R}})$$ of generalized rational functions inside the Hahn series field, and the Cauchy completion of $$\Bbbk(x^{\mathbb{R}})$$ in its valuation uniformity.

Does this field have a standard name and/or a notation? Is it an instance of some more general construction (e.g. replacing $$\mathbb{R}$$ by something more general, which would presumably then also include the Levi-Civita field as the case of $$\mathbb{Q}$$)?

• This is the completion of $\mathbb{k}(x^{\mathbb{R}})$ as a valued field, i.e. the unique up to isomorphism dense valued field extension without proper dense valued field extension. – nombre Jul 6 '19 at 10:29
• It is a field because the multiplication is well-defined commutative and the geometric series implies $1+\sum_{n=1}^\infty a_n x^{b_n}, 0 < b_n <b_{n+1} \to \infty$ has an inverse. – reuns Jul 6 '19 at 22:05

• Following Google to en.wikipedia.org/wiki/Novikov_ring and thence by an external link back to MO, I found mathoverflow.net/a/13226/49 where my ring is called the "universal Novikov field" (hence I guess it is a field!). Going back to Google, I see that this terminology seems to be common. Unfortunately the most common notation for it is $\Lambda$ or $\Lambda(\Bbbk)$, which works in the context of a particular paper but can't really be a global notation (like $\Bbbk[[x]]$) -- for one thing, it clashes with the standard notation for exterior algebras. But a name helps! – Mike Shulman Jul 6 '19 at 15:48