Real inflection points of real linear series on an elliptic curve

Given a real elliptic curve $E$ with non-empty real part and $[D]\in \mbox{Pic}^2 E$ its $g_2^1$, we study the real inflection points of distinguished subseries of the complete real linear series $|\mathcal{L}_\mathbb{R}(kD)|$ for $k\geq 3$. We define {\it key polynomials} whose roots index the ($x$-coordinates of) inflection points of the linear series, away from the points where $E$ ramifies over $\mathbb{P}^1$. These fit into a recursive hierarchy, in the same way that division polynomials index torsion points. Our study is motivated by, and complements, an analysis of how inflectionary loci vary in the degeneration of real {\it hyperelliptic} curves to a metrized complex of curves with elliptic curve components that we carried out in our previous article with Biswas.