Bulletin of the Belgian Mathematical Society - Simon Stevin
- Bull. Belg. Math. Soc. Simon Stevin
- Volume 19, Number 2 (2012), 367-372.
Zeros of the derivative of a p-adic meromorphic function and applications
Let $K$ be an algebraically closed field of characteristic 0, complete with respect to an ultrametric absolute value. We show that if the Wronskian of two entire functions in $K$ is a polynomial, then both functions are polynomials. As a consequence, if a meromorphic function $f$ on all $K$ is transcendental and has finitely many multiple poles, then $f'$ takes all values in $K$ infinitely many times. We then study applications to a meromorphic function $f$ such that $f'+bf^2$ has finitely many zeros, a problem linked to the Hayman conjecture on a p-adic field.
Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 2 (2012), 367-372.
First available in Project Euclid: 24 May 2012
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Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 12J25: Non-Archimedean valued fields [See also 30G06, 32P05, 46S10, 47S10] 46S10: Functional analysis over fields other than $R$ or $C$ or the quaternions; non-Archimedean functional analysis [See also 12J25, 32P05]
Boussaf, Kamal; Escassut, Alain; Ojeda, Jacqueline. Zeros of the derivative of a p-adic meromorphic function and applications. Bull. Belg. Math. Soc. Simon Stevin 19 (2012), no. 2, 367--372. doi:10.36045/bbms/1337864279. https://projecteuclid.org/euclid.bbms/1337864279