My question is rather philosophical : can a meromorphic function of normalized norm with simple zeros on the flat torus stay close to zero on a large set when its zeros are far from each other ?
The image I have in mind is that when two zeros collide, at the limit case, the derivative is zero. In the opposite, when your zeros are going far from each other how does the derivative at the zeros grow ? Or the L¹ norm of the inverse of the function ?
The first element I have in mind is the following example :
Let T be a flat torus of dimension 1 with the canonical metric coming from $\mathbb C$. Choose $P_1, P_2$ two distinct points on it.
Let $\Gamma (P_1 + P_2)$ be the 2 dimensional vector space of meromorphic functions of T with at most simple poles in $P_1$ and $P_2$. For such a function f, we can define a norm, by taking the integral $\int_T |f|$ according to the flat metric.
Then we look at $S_\epsilon\subset \Gamma$ the compact subset of norm 1 for which the zeros are at "distance" at least $\epsilon$. Formally, $\int_\gamma |f| \geq \epsilon$ for any path joining two zeros.
For any function in $S_\epsilon$, we can evaluate its derivative at the zeros, and its norm should be always strictly bigger than zero. Since $S_\epsilon$ is compact there is a lower bound.
Now the question is how this lower bound depend on $\epsilon$ ? What can we say on this lower bound ?
