Following work of Mazur-Tate and Satoh, we extend the definition of division polynomials to arbitrary isogenies of elliptic curves, including those whose kernels do not sum to the identity. In analogy to the classical case of division polynomials for multiplication-by-n, we demonstrate recurrence relations, identities relating to classical elliptic functions, the chain rule describing relationships between division polynomials on source and target curve, and generalizations to higher dimension (i.e., elliptic nets).

翻译：基于Mazur-Tate与Satoh的工作，我们将分裂多项式的定义扩展至椭圆曲线的任意同源，包括核元素之和非单位元的同源情形。类比经典n倍映射的分裂多项式，我们证明了递推关系、与经典椭圆函数相关的恒等式、描述源曲线和目标曲线上分裂多项式关系的链式法则，以及到高维（即椭圆网）的推广。