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：继Mazur-Tate和Satoh的研究工作，我们将除多项式的定义推广至椭圆曲线的任意同源映射，包括其核不满足恒等元求和条件的情形。类比经典情形中乘n映射的除多项式，我们证明了递推关系、与经典椭圆函数相关的恒等式、描述源曲线与目标曲线上除多项式关联的链式法则，以及向高维情形（即椭圆网）的推广。