  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 $a_i\in\mathrm
C^{\infty}([0,1])$. 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 of
the 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 $i=1,\cdots,m$. By
means of Lagrange interpolation formula, we proves that the equation has at
most $m$ isolated periodic solutions (counted with multiplicities) if
hypothesis (H) holds, and the upper bound is sharp. Furthermore, this
conclusion is also obtained under some weaker geometric hypotheses.
  Applying our main result for the trigonometrical generalized Abel equation
with coefficients of degree one, we give a criterion to obtain the upper bound
for the number of isolated periodic solutions. This criterion is "almost
equivalent" to hypothesis (H) and can be much more effectively checked.
