Is counting roots with multiplicites at all a geometric concept? It is well known that a polynomial of degree $n$ admits $n$ roots when the field is algebraically closed. However, this comes with a caveat, in particular that the roots be counted with multiplicity.
From an algebraic standpoint, counting roots with multiplicity is very natural: If a root $\alpha_i$ has multiplicity $m_i$, then we may factorize the polynomial as $a \displaystyle \prod_{i=1}^l (x - \alpha_i)^{m_i}$.
However, this is unsatisfying to me since we are first introduced to roots of a polynomial as a geometric concept: in the real case, it's places where the curve crosses the $x$-axis. Thus, when we look at the theorem from a purely geometric standpoint without looking at the underlying algebraic framework, the theorem becomes a lot less interesting: the polynomial of degree $n$ admits $n$ roots, but it can seem to admit less if some of those roots happen to have multiplicity greater than one.
My question is, is there any geometric relation between a root and its multiplicity which allow us to see the full strength of the theorem (the existence of $n$ roots) without relying on the underlying algebraic structure of the polynomial? Stated differently, can we look at the graph/image of a polynomial and determine how many roots it has (without appealing to arguments about its degree and inferring the number of roots from that)?
Note that my question can also be applied to something like Bezout's theorem where plane curves of degree $m$, $n$ intersect $mn$ times, assuming the intersections are counted with multiplicity. The condition that they be counted with multiplicity is even more disappointing to the geometric nature here.
 A: Consider function $f(z)=z^n$. As you go around a small contour around $z=0$ which is $n$-fold zero of $f$, $f(z)$ "goes around" $0$ in complex plane $n$ times. The same will be true for any holomorphic function with $n$-fold zero at some point (meaning that it is zero there together with its first $n-1$ derivatives). This multiplicity of zeros is therefore connected to topological invariants such as winding number. It is also connected with notion of degree of a mapping, which I will try to explain analysing example of polynomials in complex plane. If you have $n$-th order polynomial $g(z)$ and go with $z$ around a huge contour which contains all the zeros of $g$ inside then $g(z)$ will wrap around complex plane $n$ times. It can be shown that functions with this property (which needs to be more rigorously defined but I'm just trying to give you the idea) must have $n$ zeros counting with multiplicities. Fundamental theorem of algebra follows from this. It is crucial that you count with multiplicities for the following reason: degree of a mapping tells you how much you "wrap around in total". Multiplicity of a zero tells you "how much you wrap around when you go around this zero". Therefore sum of multiplicities must be equal to degree. For polynomials degree is the same as order of polynomial. But idea of a degree is much more general and it applies to continuous function between compact spaces. Complex plane itself is not complact but it can be made compact by adding one "point at infinity", promoting it to the celebrated Riemann sphere.
