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. $Z(f)=\emptyset$)