# A counterexample to the composition condition conjecture for polynomial Abel differential equations

## TL;DR

This paper presents a counterexample to the long-standing conjecture that all centers of polynomial Abel differential equations satisfy the composition condition, challenging previous assumptions in the field.

## Contribution

The authors provide the first known counterexample to the conjecture that all polynomial Abel differential equation centers meet the composition condition.

## Key findings

- Counterexample disproves the conjecture
- Not all polynomial Abel centers satisfy the composition condition
- Challenges existing beliefs in the theory of polynomial Abel equations

## Abstract

The Polynomial Abel differential equations are considered a model problem for the classical Poincar\'e center--focus problem for planar polynomial systems of ordinary differential equations. Last decades several works pointed out that all the centers of the polynomial Abel differential equations satisfied the composition conditions (also called universal centers). In this work we provide a simple counterexample to this conjecture.