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This paper is devoted to the investigation of generalized Abel equation$\dot{x}=S(x,t)=\sum^m_{i=0}a_i(t)x^i$, where $a_i\in\mathrmC^{\infty}([0,1])$. A solution $x(t)$ is called a {\em periodic solution} if$x(0)=x(1)$. In order to estimate the number of isolated periodic solutions ofthe equation, we propose a hypothesis (H) which is only concerned with $S(x,t)$on $m$ straight lines: There exist $m$ real numbers$\lambda_1<\cdots<\lambda_m$ such that either $(-1)^i\cdot S(\lambda_i,t)\geq0$for $i=1,\cdots,m$, or $(-1)^i\cdot S(\lambda_i,t)\leq0$ for $i=1,\cdots,m$. Bymeans of Lagrange interpolation formula, we proves that the equation has atmost $m$ isolated periodic solutions (counted with multiplicities) ifhypothesis (H) holds, and the upper bound is sharp. Furthermore, thisconclusion is also obtained under some weaker geometric hypotheses.  Applying our main result for the trigonometrical generalized Abel equationwith coefficients of degree one, we give a criterion to obtain the upper boundfor the number of isolated periodic solutions. This criterion is "almostequivalent" to hypothesis (H) and can be much more effectively checked.

A geometric criterion for equation $\dot{x}=\sum^m_{i=0}a_i(t)x^i$ having at most $m$ isolated periodic solutions