Let $\pi:E\to M$ be a smooth vector bundle over a smooth manifold, with $\text{rank}(E)=\text{dim}(M)$. For a section $\sigma$ of $E$ with a zero at $p\in M$, define the degree of the zero at $p$ to be the topological degree of the induced map from a small sphere in $T_pM$ to a small sphere in $E_p$.
One motivation for studying degrees of zeros is that they contain information about the topology of $E$. I think the following is true, although I couldn't find a good reference:
Theorem 1 (Hopf index theorem). Suppose the zeroes of $\sigma$ are the isolated points $p_1, \ldots p_k$, with degrees $d_1,\ldots d_k$ respectively. Then the Euler class of $E$ is $\chi(E)=\sum_{i=1}^kd_i$.
With this as motivation, my first question, the one stated in the title, is roughly (see the Example for an idea of what I'm getting at, and feel free to suggest a sharper version):
Are there conditions on [say, the symbol of] a linear differential operator $D:E\to F$, such that [some constraint] is satisfied by degree of any zero $p\in M$ of any local solution $\sigma\in\Gamma(E)$ to the PDE $D\sigma=0$?
Example. If $M$ is a Riemann surface and $E$ a holomorphic line bundle over it, the kernel of the delbar operator $\overline{\partial}:E\to T^{0,1}M\otimes E$ is precisely the holomorphic sections of $E$. By complex analysis, zeroes of holomorphic functions have positive degree.
Theorem 1 then yields the standard result that if a line bundle admits a global holomorphic section then its Euler class (aka first Chern class) is nonnegative.
Here's an idea I had for trying to prove a theorem of the sort I ask for in Question 1. Recall the definition of the local ring of a zero $p\in M$ of a section of E:
Write $\mathcal{O}_p$ for the ring of germs of smooth functions about $p$.
Definition. Let $\sigma\in\Gamma(E)$ be a smooth section which vanishes at $p$. The local ring of the germ $[\sigma]_p$, denoted $Q([\sigma]_p)$, is the quotient $\mathcal{O}_p/([\sigma]_p)$, where $([\sigma]_p)$ is the ideal of $\mathcal{O}_p$ generated by "components of $\sigma$": $([\sigma]_p)=\ <\{[v(\sigma)]_p:v\text{ a nonvanishing section of }E^*\}> \ \subseteq \mathcal{O}_p$.
Theorem 2 (Eisenbud-Levine-Khimshiashvili). Suppose $p$ is a zero of $\sigma$, and the local ring $Q([\sigma]_p)$ is a finite-dimensional algebra over $\mathbb{R}$. Then there is a canonical quadratic form on $Q([\sigma]_p)$, such that the degree of the zero of $\sigma$ at $p$ can be calculated as this quadratic form's signature.
Because a system of PDE is precisely a constraint on the local behaviour of a section, it seems plausible that local rings of zeros of solutions of a PDE might have interesting properties.
Are there conditions on [say, the symbol of] a linear differential operator $D:E\to F$, such that [some constraint] is satisfied by the signature of the local ring $Q([\sigma]_p)$ of any zero $p\in M$ of any local solution $\sigma\in\Gamma(E)$ to the PDE $D\sigma=0$?
Example. As in the previous example, let $E$ be a holomorphic line bundle over a Riemann surface $M$. By manipulating the Cauchy-Riemann equations, one can (I think!) classify the possible local rings of zeroes of a holomorphic section, and show that all of them have positive signature.
Theorem 2 then yields an alternative proof of the quoted result that zeroes of holomorphic functions have positive degree.
