Given $P(x,y)=0$, where $P(x,y)$ is a polynomial,and we get the function $y=f(x),P(x,y)=0$, then what is the sufficient condition for the existence of poles of algebraic function $f(x)$? what is the sufficient and necessary condition for poles of $f(x)$, if it is possible to give necessary condition.
Furthermore, could we know the number of poles by any algorithm?
If I'm understanding correctly, your question boils down to the following: take the closure of the curve described by $P(x,y)$ (which we think of as "the graph of $y = f(x)$") in $\mathbb{A}^1_{x} \times \mathbb{P}^1_y$, and intersect it with the $y = \infty$ line.
Algebraically, this boils down to writing $P(x,y) = g_0(x) + g_1(x) y + \cdots + g_n(x)y^n$, and looking for the zeros of $g_n(x)$.
(Formally, we would homogenize $P(x,y)$ with respect to $y$ only, obtaining $\tilde P = g_0(x) Y_1^n + g_1(x)Y_0 Y_1^{n-1} + \cdots + g_n(x) Y_0^n$. Then we intersect with the line $y=\infty$ by setting $Y_1 = 0, Y_0 = 1$, which kills everything except $g_n(x)$.)
Note: if you are also interested in the behavior of $y = f(x)$ as $x\to \infty$, then conduct the above procedure in $\mathbb{P}^1_{x} \times \mathbb{P}^1_y$ instead, i.e. homogenize $g_n(x)$ and consider what happens to it as $x \to \infty$.