Optimal Curve Straightening is $\exists\mathbb{R}$-Complete

We prove that the following problem has the same computational complexity as the existential theory of the reals: Given a generic self-intersecting closed curve $\gamma$ in the plane and an integer $m$, is there a polygon with $m$ vertices that is isotopic to $\gamma$? Our reduction implies implies two stronger results, as corollaries of similar results for pseudoline arrangements. First, there are isotopy classes in which every $m$-gon with integer coordinates requires $2^{\Omega(m)}$ bits of precision. Second, for any semi-algebraic set $V$, there is an integer $m$ and a closed curve $\gamma$ such that the space of all $m$-gons isotopic to $\gamma$ is homotopy equivalent to $V$.