Say $L\mathbb{C}^\times$ is the loop group of smooth maps $S^1 \to \mathbb{C}^\times$. There is a submonoid $L_{poly}\mathbb{C}^\times$ of loops that look like $w_0 + w_1z +w_2z^2 + \cdots + w_nz^n$ where $z = e^{i\theta}$ (as Andrew notes below this is not a group because its not closed under taking inverses). Equivalently $L_{poly}\mathbb{C}^\times$ as a set is just polynomials $p(z) \in \mathbb{C}[z]$ such that $p(z) \ne 0$ for $|z| = 1$.

If we mod out by scaling and rotation then the set of polynomial loops describe a subset $X$ of $\mathbb{P}(\oplus_{n \in \mathbb{N}} \mathbb{C})$ (by identifying a loop with its vector of coefficients). I want to look at $X$ from an algebro-geometric point of view, but I have no intuition about how bad or nice $X$ may be; i.e. can it be a variety?

The way I've been thinking about it is that $X = \cup_{n \in \mathbb{N}} X_n$ where $X_n \subset \mathbb{P}^n$ are the loops of degree at most $n$. I think $X_1$ is the image under the projection $\mathbb{C}^2 - 0 \to \mathbb{P}^1$ of the set {$(w_0,w_1):|w_0|\ne |w_1|$}. So it seems describable as the complement of a hypersurface in $\mathbb{R}^4$ but probably its not a complex variety.

But already trying to figure out what $X_2$ is seems difficult. Also I feel I don't have any `sophisticated' way of thinking about this stuff meaning my attempts to describe $X_2$ seems to always degenerate to just fumbling around with planar geometry.

Some specific questions regarding this setup:

0) What is the dimension of $X_n$?

1) Which if any of the $X_n$ or $X$ are a variety over $\mathbb{C}$ or $\mathbb{R}$?

2) If $X$ or $X_n$ are not varieties can you find any positive dimensional varieties contained in them?

3) Can you suggest any tools that might be useful for answering any of the previous questions?

Of course if any of this seems to easy you are welcome to replace $\mathbb{C}^\times$ with $GL(n,\mathbb{C})$, polynomials with rational functions or with power series convergent in an annulus containing $|z| = 1$.

An extra thought: So $L_{poly}\mathbb{C}^\times$ is not a group, but it seems you do get a group if you look at convergent series in non positive powers of $z$; i.e. loops that look like $\sum_{j \in \mathbb{N}} c_j z^{-j}$. I wonder if there's anything interesting you can say about the $c_j$.