# Center conditions III: Parametric and model center problems

## Authors

- Received:

DOI: 10.1007/BF02803517

- Cite this article as:
- Briskin, M., Francoise, J.-. & Yomdin, Y. Isr. J. Math. (2000) 118: 83. doi:10.1007/BF02803517

## Abstract

We consider an Abel equation (*)*y’*=*p(x)y*
^{2} +*q(x)y*
^{3} with*p(x), q(x)* polynomials in*x*. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane) is that*y*
_{0}=*y*(0)≡*y*(1) for any solution*y(x)* of (*).

Folowing [7], we consider a parametric version of this condition: an equation (**)*y’*=*p(x)y*
^{2} +*εq(x)y*
^{3}
*p, q* as above, ε ∈ ℂ, is said to have a parametric center, if for any ɛ and for any solution*y*(ɛ,*x*) of (**)*y*(ɛ, 0)≡*y*(ɛ, 1)..

We give another proof of the fact, shown in [6], that the parametric center condition implies vanishing of all the moments*m*
_{
k
} (1), where*m*
_{
k
}(*x*)=∫
_{0}
^{
x
}
^{
pk
}(*t*)*q*(*t*)(*dt*),*P*(*x*)=∫
_{0}
^{
x
}
*p(t)dt*. We investigate the structure of zeroes of*m*
_{
k
}
*(x)* and generalize a “canonical representation” of*m*
_{
k
}
*(x)* given in [7]. On this base we prove in some additional cases a composition conjecture, stated in [6, 7] for a parametric center problem.