K-rational points on curves

Mazur and Rubin's ``Diophantine stability'' program suggests asking, for a given curve $C$, over what fields $K$  does $C$ have rational points, or at least to study the degrees of such $K$. We study this question for planar curves $C$ from various perspectives and relate solvability to the shape of $C$'s Newton polygon (the real original one that Newton worked with, not a $p$-adic one which are  frequently used in arithmetic geometry research). This is joint work with Lea Beneish.