On the fundamental groups of non-generic $\mathbb{R}$-join-type curves, II

Abstract

We study the fundamental groups of (the complements of) plane complex curves defined by equations of the form $f(y)=g(x)$, where $f$ and $g$ are polynomials with real coefficients and real roots (so-called $\mathbb{R}$-join-type curves). For generic (respectively, semi-generic) such polynomials, the groups in question are already considered in [6] (respectively, in [3]). In the present paper, we compute the fundamental groups of $\mathbb{R}$-join-type curves under a simple arithmetic condition on the multiplicities of the roots of $f$ and $g$ without assuming any (semi-)genericity condition.