This paper is devoted to the investigation of Abel equation x˙=S(t,x)=∑i=03ai(t)xi\documentclass[12pt]{minimal}
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\begin{document}$$\dot{x}=S(t,x)=\sum ^3_{i=0}a_i(t)x^i$$\end{document}, where ai∈C∞([0,1])\documentclass[12pt]{minimal}
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\begin{document}$$a_i\in \mathrm C^{\infty }([0,1])$$\end{document}. A solution x(t) with x(0)=x(1)\documentclass[12pt]{minimal}
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\begin{document}$$x(0)=x(1)$$\end{document} is called a periodic solution. And an orbit x=x(t)\documentclass[12pt]{minimal}
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\begin{document}$$x=x(t)$$\end{document} is called a limit cycle if x(t) is a isolated periodic solution. By means of Lagrange interpolation formula, we give a criterion to estimate the number of limit cycles of the equation. This criterion is only concerned with S(t, x) on three non-intersecting curves. Applying our main result, we prove that the maximum number of limit cycles of the equation is 4 if a2(t)a0(t)<0\documentclass[12pt]{minimal}
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\begin{document}$$a_2(t)a_0(t)<0$$\end{document}. To the best of our knowledge, this is a nontrivial supplement for a classical result which says that the equation has at most 3 limit cycles when a2(t)≠0\documentclass[12pt]{minimal}
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\begin{document}$$a_2(t)\ne 0$$\end{document} and a0(t)≡0\documentclass[12pt]{minimal}
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\begin{document}$$a_0(t)\equiv 0$$\end{document}. We also study a planar polynomial system with homogeneous nonlinearities: x˙=ax-y+Pn(x,y),y˙=x+ay+Qn(x,y),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \dot{x}=ax-y+P_n(x,y),\ \ \dot{y}=x+ay+Q_n(x,y), \end{aligned}$$\end{document}where a∈R\documentclass[12pt]{minimal}
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\begin{document}$$a\in \mathbb R$$\end{document} and Pn,Qn\documentclass[12pt]{minimal}
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\begin{document}$$P_n, Q_n$$\end{document} are homogeneous polynomials of degree n≥2\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 2$$\end{document}. Denote by ψ(θ)=cos(θ)·Qn(cos(θ),sin(θ))-sin(θ)·Pn(cos(θ),sin(θ))\documentclass[12pt]{minimal}
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\begin{document}$$\psi (\theta )=\cos (\theta )\cdot Q_n\big (\cos (\theta ),\sin (\theta )\big )-\sin (\theta )\cdot P_n\big (\cos (\theta ),\sin (\theta )\big )$$\end{document}. We prove that if (n-1)aψ(θ)+ψ˙(θ)≠0\documentclass[12pt]{minimal}
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\begin{document}$$(n-1)a\psi (\theta )+\dot{\psi }(\theta )\ne 0$$\end{document}, then the polynomial system has at most 1 limit cycle surrounding the origin, and the multiplicity is no more than 2.
