Abstract

Given a real elliptic curve E with non-empty real part and a real effective divisor $\mathcal{D}$ on E arising via pullback from ${\mathbb{P}}^1$ under the hyperelliptic structure map, we study the real inflection points of distinguished subseries of the complete real linear series $\left|\mathcal{D}\right|$ on E. We define inflection 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 hyperelliptic curves to a metrized complex of curves with elliptic curve components that we carried out in an earlier joint work with I. Biswas.

摘要

给定具有非空实部和实有效除数的实椭圆曲线E$\mathcal{D}$在E上通过回调产生${\mathbb{磷}}^1$在超椭圆结构映射下，我们研究了完整实线性级数的可区分子级数的实拐点$\left|\mathcal{D}\right|$在E上。我们定义了拐点多项式，其根索引线性级数的拐点（的x坐标），远离E分支的点${\mathbb{磷}}^1$. 这些符合递归层次结构，与除法多项式索引扭转点的方式相同。我们的研究的动机和补充是，分析了在我们与 I. Biswas 的早期联合工作中进行的真实超椭圆曲线退化为具有椭圆曲线分量的曲线的度量化复合体时，屈折位点如何变化的分析。