Explicit Implicit Function Theorem for All Fields

TL;DR

This paper presents an explicit implicit function theorem applicable to formal power series over all fields, generalizing classical formulas like Lagrange inversion and Flajolet-Soria coefficient extraction.

Contribution

It introduces a universal implicit function theorem for formal power series valid across all fields, extending known formulas beyond characteristic zero.

Findings

Generalized Lagrange inversion formula for all fields

Derived Flajolet-Soria coefficient extraction formula for all fields

Unified approach to implicit functions in formal power series

Abstract

We give an explicit implicit function theorem for formal power series that is valid for all fields, which implies in particular Lagrange inversion formula and and Flajolet-Soria coefficient extraction formula known for fields of characteristic 0.