I have been reading some stuff about Riemann surfaces and in the book , " Algebraic Curves and Riemann surfaces" he talks about the multiplicty of $F$ at $p$. I understand the definition but i cant quite get some intuition behind it at first i though it was the number of pre-images of the point but that is wrong so i dont know how to go on to think about . For example for the function $z^m$ between two disks in the complex plane the multiplicity for $m \neq 0$ is $1$ and for $0$ is $m$.
Same thing for the degree of a holomorphic map between compact riemann surfaces. So if anyone could give me some more insight and intuition about the definitions i would be very thankfull. Thanks in advance.
The degree counts the number $d$ of pre-images of a general point. The multiplicity counts how this fails for "non-general" points so that the pre-image of any point has $d$ points taking account of the multiplicities.
Consider the map from the Riemann sphere to itself $p\colon \overline{\mathbb{C}}\rightarrow \overline{\mathbb{C}}$ given by $$p(z) = z^3(z-1)^2.$$ Since it is a degree $5$ polynomial, for $t\in \mathbb{C}\setminus \left\{0, \frac{108}{3125}\right\}$ $$ p^{-1}(t) = \{z \mid z^3(z-1)^2 = t \} $$ has five distinct zeros, hence the degree of $p$ is five. We also have that $p^{-1}(\infty) = \{\infty\}$ with multiplicity $5$ -- in a neighborhood of the infinity $p(z) = \dfrac{z^5}{(1-z)^2}$ -- and $p^{-1}(0) = \{0,1\}$ where $0$ has multiplicity 3 and $1$ has multiplicity $2$. I won't go into detail but $p^{-1}\left(\frac{108}{3125} \right)$ contains $\frac{3}{5}$ with multiplicity $2$ and other $3$ simple points.