On the roots of a \(p\)-adic rational function (Q2751742)

Let \(p\) be a prime number greater than \(2\), and let \(K\) be the field of \({\mathbb C}_p\). Let \(K^0[X]=\{P \in K[X]\); \(P(0)=0\}\) and \(A^0[X]=\{f \in A(D)\); \(f(0)=0\}\), where \(D\) is the unit disk centered at \(0\), and \(A(D)\) is the algebra of the bounded power series converging in \(D\). This paper studies the following conjectures Conjectures \(C_j\): Let \(j\geq 1\) be an integer. If there exist \(P, Q \in K^0[X]\) and \(U \in A^0[X]\) such that for every \(x\in D\), NEWLINE\[NEWLINE{1+P(x)\over 1+Q(x)} = (1+U(x))^{p^j}NEWLINE\]NEWLINE then there exists \(A, B \in K^0[X]\) and \(\varphi \in A^0[X]\) with \(\|\varphi\|\leq p^{-{1\over p-1}}\) and such that, for every \(x \in D\), NEWLINE\[NEWLINE1+U(x)=\left({1+A(x)\over 1+B(x)}\right)(1+\varphi(x)).NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEThe conjecture \(C_1\) was proved by the authors. In this paper, it is shown that the conjecture \(C_j\) may be proved by a recurrent proof, i.e., let \(J>2\) be an integer, if Conjecture \((C_j)\) is true for all \(j\) less than \(J\), then Conjecture \((C_J)\) is true.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00058].