Branched values and quasi-exceptional values for \(p\)-adic meromorphic functions (Q2862649)

Let \(K\) be an algebraically closed field of characteristic \(0\) that is complete with respect to an ultrametric absolute value, with residue characteristic \(p\). Let \(f\) be a meromorphic function on \(K\) or on an ``open'' disk in \(K\) and let \(b\in K\) be given. Then \(b\) is said to be a perfectly branched value for \(f\) if all but finitely many of the zeros of \(f-b\) are multiple zeros, \(b\) is said to be a totally branched value for \(f\) if all the zeros of \(f-b\) are multiple zeros, and \(b\) is said to be a quasi-exceptional value of \(f\) if \(f-b\) has only finitely many zeros. In this paper the authors state and prove results about the maximum number of perfectly branched values, totally branched values or quasi-exceptional values that a meromorphic function can have, which are similar to the corresponding results from classical Complex Analysis. More specifically, the authors prove the following results:NEWLINENEWLINE1) Any meromorphic function on \(K\) has at most three totally branched values.NEWLINENEWLINE2) If \(f\) is a transcendental meromorphic function on \(K\) or an unbounded meromorphic function on an open disk in \(K\) then \(f\) has at most four perfectly branched values. If \(f\) is a transcendental meromorphic function on \(K\) that has finitely many poles then \(f\) has at most one perfectly branched value. If \(f\) is an unbounded meromorphic function on an open disk in \(K\) that has finitely many poles then \(f\) has at most two perfectly branched values; moreover, if \(p=0\) then \(f\) has at most one perfectly branched value.NEWLINENEWLINE3) If \(P\) is a polynomial with coefficients in \(K\) then \(P\) has at most one totally branched value.NEWLINENEWLINE4) If \(f\) is a transcendental meromorphic function on \(K\) which has \(0\) and \(\infty\) as perfectly branched values then \(f\) has no nonzero quasi-exceptional values; that is, \(f\) assumes all nonzero values infinitely often. This is an improvement of an earlier result by K. Boussaf and J. Ojeda published in a different paper.NEWLINENEWLINE5) If \(p\neq2\) and if \(f\) is an unbounded meromorphic function on an open disk in \(K\) which has all but finitely many zeros and poles of even order, then \(f\) has no nonzero quasi-exceptional values.