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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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