  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.
