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On the zeros of the Wronskian of an entire or meromorphic function and its derivatives. (English) Zbl 0865.30044
Let $$g$$ be a meromorphic function in the complex plane, and define the homogeneous differential polynomial $$\psi$$ by $$\psi= W(g, g^{(k_1)}, g^{(k_2)}, \dots, g^{(k_{n-1})})$$ where $$W$$ denotes the Wronskian and $$k_1, k_2, \dots, k_{n-1}$$ are pairwise distinct positive integers.
In the case of an entire function $$g$$, we give sharp upper and lower bounds for the Nevalinna counting function $$N(r, 1/ \psi)$$ of the zeros of $$\psi$$ in terms of $$N(r, 1/g)$$. In particular, we show that if $$g$$ is not an exponential sum then $$\psi$$ has few zeros in the sense that $$N(r, 1/\psi)= S(r, g)$$ if and only if $$N(r, 1/g)= S(r,g)$$. One of the main tools is a new result on the proximity function of quotients of certain Wronskians which might be of independent interest.
For meromorphic functions $$g$$, we present two methods to obtain lower bounds for $$N(r,1/ \psi)$$ in terms of $$N(r,1/g)$$ and $$\overline {N} (r,g)$$. As a tool, we give formulas for the coefficients of the greatest common divisor of two linear differential operators.

MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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