A note on algorithmic approach to inverting formal power series
Elżbieta Adamus
TL;DR
The paper extends an existing polynomial inversion algorithm to inverting formal power series over fields of arbitrary characteristic by recasting the problem in the $I$-adic topology and using an alternating-sum construction $G=\sum_{l}(-1)^l P_l$ built from the operator $\Delta_F$. It proves convergence to the formal inverse for maps $F$ with $F(0)=0$ and $F(X)=X+H(X)$ where $\operatorname{ord}(H_i)>1$, and provides explicit truncation bounds $D$ and $\mu_i$ that allow finite computation of the degree-$D$ part of $G$ even when the inverse is not Pascal finite. When the inverse is a polynomial of degree $D$, the paper establishes a symmetry-based decomposition that yields $G$ from $F$ by finite truncations using $\widetilde{P_k^i}^D$ and related $Q_s$, $R_\mu$ constructs. Through one- and two-variable examples, including a case with $G(Y)=Y+Y^2$ and a trigonometric two-variable map, the work demonstrates how the method computes inverses with controlled precision and discusses stabilization of coefficients in the power series expansion. This provides a practical, characteristic-agnostic framework for constructive inverse computation in formal power series.
Abstract
In our previous paper an effective algorithm for inverting polynomial automorphisms was proposed. We extend its application to the case of formal power series over a field of arbitrary characteristic and illustrate the proposed approach with some examples.