# The Faa di Bruno Hopf algebra for multivariable feedback recursions in the center problem for higher order Abel equations

## TL;DR

This paper introduces a multivariable Hopf algebra framework to analyze the center problem for higher order Abel equations, linking feedback control, combinatorial algebra, and differential equations.

## Contribution

It generalizes the Faa di Bruno Hopf algebra to multivariable feedback recursions, providing new algebraic tools for the center problem in higher order Abel equations.

## Key findings

- Develops a multivariable Hopf algebra for feedback recursions.
- Provides a linear recursion for the antipode using coderivations.
- Explores the composition condition for the center problem.

## Abstract

Poincare's center problem asks for conditions under which a planar polynomial system of ordinary differential equations has a center. It is well understood that the Abel equation naturally describes the problem in a convenient coordinate system. In 1989, Devlin described an algebraic approach for constructing sufficient conditions for a center using a linear recursion for the generating series of the solution to the Abel equation. Subsequent work by the authors linked this recursion to feedback structures in control theory and combinatorial Hopf algebras, but only for the lowest degree case. The present work introduces what turns out to be the nontrivial multivariable generalization of this connection between the center problem, feedback control, and combinatorial Hopf algebras. Once the picture is completed, it is possible to provide generalizations of some known identities involving the Abel generating series. A linear recursion for the antipode of this new Hopf algebra is also developed using coderivations. Finally, the results are used to further explore what is called the composition condition for the center problem.