The Abel differential equation y′=p(x)y3+q(x)y2\documentclass[12pt]{minimal}
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\begin{document}$$y'=p(x)y^3 + q(x) y^2$$\end{document} with meromorphic coefficients p,q\documentclass[12pt]{minimal}
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\begin{document}$$p,q$$\end{document} is said to have a center on [a,b]\documentclass[12pt]{minimal}
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\begin{document}$$[a,b]$$\end{document} if all its solutions, with the initial value y(a)\documentclass[12pt]{minimal}
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\begin{document}$$y(a)$$\end{document} small enough, satisfy the condition y(a)=y(b)\documentclass[12pt]{minimal}
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\begin{document}$$y(a)=y(b)$$\end{document}. The problem of giving conditions on (p,q,a,b)\documentclass[12pt]{minimal}
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\begin{document}$$(p,q,a,b)$$\end{document} implying a center for the Abel equation is analogous to the classical Poincaré Center-Focus problem for plane vector fields. Following Briskin et al. (Ergodic Theory Dyn Syst 19(5):1201–1220, 1999; Isr J Math 118:61–82, 2000); Cima et al. (Qual Theory Dyn Syst 11(1):19–37, 2012; J Math Anal Appl 398(2):477–486, 2013) we say that Abel equation has a “parametric center” if for each ϵ∈C\documentclass[12pt]{minimal}
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\begin{document}$$\epsilon \in \mathbb C$$\end{document} the equation y′=p(x)y3+ϵq(x)y2\documentclass[12pt]{minimal}
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\begin{document}$$y'=p(x)y^3 + \epsilon q(x) y^2$$\end{document} has a center. In the present paper we use recent results of Briskin et al. (Algebraic Geometry of the Center-Focus problem for Abel differential equations, arXiv:1211.1296, 2012); Pakovich (Comp Math 149:705–728, 2013) to show show that for a polynomial Abel equation parametric center implies strong “composition” restriction on p\documentclass[12pt]{minimal}
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\begin{document}$$p$$\end{document} and q\documentclass[12pt]{minimal}
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\begin{document}$$q$$\end{document}. In particular, we show that for degp,q≤10\documentclass[12pt]{minimal}
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\begin{document}$$\deg p,q \le 10$$\end{document} parametric center is equivalent to the so-called “Composition Condition” (CC) (Alwash and Lloyd in Proc R Soc Edinburgh 105A:129–152, 1987; Briskin et al. Ergodic Theory Dyn Syst 19(5):1201–1220, 1999) on p,q\documentclass[12pt]{minimal}
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\begin{document}$$p,q$$\end{document}. Second, we study trigonometric Abel equation, and provide a series of examples, generalizing a recent remarkable example given in Cima et al. (Qual Theory Dyn Syst 11(1):19–37, 2012), where certain moments of p,q\documentclass[12pt]{minimal}
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\begin{document}$$p,q$$\end{document} vanish while (CC) is violated.
