Zeroes of global sections killed by differential operators I asked this question some two weeks ago on StackExchange, but received no feedback of any sort ...

Let $X$ be a compact connected Riemann surface and let $\Phi:M\rightarrow N$ be an elliptic differential operator where $M$ and $N$ are two complex line bundles on $X$.
Let $f$ be a $C^\infty$-global section of $M$ (meaning that locally on $X^{\rm an}$, the analytic variety associated to $X$, $f$ is given by some $C^\infty$-function) such that $\Phi(f)=0$.
Assuming that $f$ has isolated zeroes, the set 
$$
Z(f)=\{x\in X\mid f(x)=0\}
$$
is finite, by compactness. Is there a uniform way to estimate the size of the set $Z(f)$?
What I mean is an inequality
$$
\mid Z(f)\mid\leq C(X,\Phi)
$$
where the constant $C(X,\Phi)$ depends, loosely speaking, only on $X$ and $\Phi$ but not on $f$.
Note 1. Since $f$ is homotopically equivalent to the $0$-section of $M$, the degree of the divisor $Z(f)$ is fixed and depends only on the topology, but in the degree some zeroes appear with a negative sign ($f$ is not holomorphic in general)
Note 2. This would be a (vast) generalization of the fact that a non-zero global holomorphic function on $X$ is constant (i.e. $Z(f)=\emptyset$)
 A: Here is a more general result. Fix  a natural number $n$.$\newcommand{\bR}{\mathbb{R}}$  Denote by $B_r$ the ball of radius $r$ centered at $0$ in $\bR^n$.  Fix a  finite dimensional vector   space $\newcommand{\eF}{\mathscr{F}}$ $\eF$ of real analytic maps  $F: B_2\to\bR^n$  with the property that  for any $F\in \eF$  the set
$$ Z_1(F):= F^{-1}(0)\cap \bar{B}_1  $$
is finite, where $\bar{B}_1$ denotes the closure of $B_1$.  I claim,that
$$\sup_{F\in\eF} \# Z_1(F)<\infty. $$
To see this,  set $N:=\dim \eF$ and  fix a basis $F_1,\dotsc, F_N$ of $\eF$. For $c\in\bR^N$ we set
$$F_c:=c_1F_1+\cdots +c_N F_N\in \eF. $$
Consider the incidence  set $\newcommand{\eZ}{\mathscr{Z}}$
$$\eZ:=\Bigl\{\; (x,c)\in \bR^n\times \bR^N;\;\;\Vert x\Vert \leq 1,\;\;F_c(x)=0\;\Bigr\}. $$ 
The set  $\eZ$ is  globally subanalytic and  the canonical  projection
$$\eZ\ni (x,c) \mapsto c\in\bR^N $$
is subanalytic and  has finite fibers: the fiber $\pi^{-1}(c)$ can be identified with the zero set $Z_1(F_c)$.
The globally subanalytic  sets  and maps form a  a special case of  tame category; see e.g. 

L. van den Dries: Tame Topology and $O$-minimal Structures,  London Math, Soc. Lect. Notes, vol 248, Cambridge University Press, 1998

(For a very fast survey of  $o$-minimality see sec. 1 of this paper.) A general result in the theory of $o$-minimal structures shows  that
$$\sup_{c\in\bR^N} \# \pi^{-1}(c) <\infty. $$
If your  line bundles are real analytic and so are the coefficients  of the operator $\Phi$ then the upper estimate that you seek reduces to the above fact in the special case $n=2$.
