Computing Reciprocals of Bivariate Power Series

Abstract

We consider the multiplicative complexity of the inversion and division of bivariate power series modulo the “triangular” ideal generated by all monomials of total degree n + 1. For inversion, we obtain a lower bound of 7/8n ² – O(n) opposed to an upper bound of 7/3n ² + O(n). The former bound holds for all fields with characteristic distinct from two while the latter is valid over fields of characteristic zero that contain all roots of unity (like e.g.ℂ ). Regarding division, we prove a lower bound of 5/4 n ² – O(n) and an upper bound of 3 5/6 n ² + O(n). Here, the former bound is proven for arbitrary fields whereas the latter bound holds for fields of characteristic zero that contain all roots of unity.

Similar results are obtained for inversion and division modulo the “rectangular” ideal (Xⁿ⁺¹,Yⁿ⁺¹).