I'm a physicist who is trying to make sense of the relationship between the number of zeros of a section from an associated vector bundle and the Euler characteristic. My interest lies in applications to gauge theories in which finite energy solutions can be classified according to a topological charge (which is a Chern number/topological degree in the cases I know, and can be related to the Euler characteristic). More precisely, let us consider theorem 11.17 in the book Differential forms ins algebraic Topology from Bott-Tu:
Let $\pi:E\to M$ be an oriented rank k vector bundle over a compact oriented manifold of dimension k. Let s be a section of E with a finite number of zeros. The Euler class of E is Poincare dual to the zeros of s, counted with the appropriate multiplicities.
I have sometimes found it stated that the topological degree counts the number of zeros of a section, with multiplicity. I believe the above theorem is the mathematical justification for this (do correct me if I am wrong, please). But the multiplicity this theorem is talking about is "the local degree of x as a singularity of the section $s/||s||$ of the unit sphere bundle of E relative to some Riemannian structure on E", according to the authors. I wanna know when (if ever) there exists a relationship between this meaning of multiplicity and that found in, say, complex analysis (that's what I called algebraic multiplicity in the title). I will assume the zeros are isolated.
My question was motivated by the behavior of axially symmetric magnetic vortices in the static case of the Nielsen-Olesen/Ginzburg-Landau theory. Here we have a scalar field $\varphi:\mathbb{R^2}\to\mathbb{C}$, seen as a section of the line bundle. Axially symmetric solutions can be taken in the form $\varphi=f(r)e^{in\theta}$, where $n$ is the (integer) topological degree. This field is coupled to a connection $A_{\theta}=A_{\theta}(r)$. The boundary conditions ensure those fields are nonsingular and that the magnetic flux is proportional to $n$. $\varphi$ must have a zero at the origin (and nowhere else). The multiplicity of this zero, in the sense described by Bott-Tu, is indeed $n$, and it may be verified that $f(r)\propto r^n$ to leading order in its power series expansion, so $f(r)$ has a zero of multiplicity $n$. This sounds a lot like the Argument Principle, with the difference that $\varphi$ does not have a complex domain. If $r$ and $\theta$ could be seen as polar coordinates in the complex plane, then this would be a zero of multiplicity $n$ in $\mathbb{C}$. The exact same $r^n$ behavior appears in all vortex theories I know, like [Maxwell or pure] Chern-Simons and many other generalized models, some very different from Nielsen-Olesen.
I'm interested in developing models in gauge theories such as the aforementioned ones (with the exact same topology, bundle and covariant derivative, but different equations of motion), and would like to know if I should expect such a behavior to occur in the solutions to such theories.
Could I find a solution where, for example, $f(r)\propto r^m$, where $m\neq n$ or would that somehow lead to a problem in my theory? Can the degree be used to predict anything about the multiplicity of the zeros of $f(r)$?
Edit (because I accidentally pressed enter before finishing the current bounty description, and didn't find a way to edit that description): While a very good answer has been provided, I still haven't been able to figure out (even after reading some of the references) if something like $f(r)\propto r^m$ could be obtained as a leading order approximation near the origin, with $m\neq n$. The answer to that might be implicit from the current answer, but I can't see it. A definite answer to the last paragraph preceding this edit is sufficient for the reward (although any information concerning sections with a form more general than the proposed $f(r) e^{in\theta}$ will be greatly appreciated as well).