Let $(K,\nu )$ be a valued field, where $\nu$ is a rank one discrete valuation. Let $R$ be its ring of valuation, $𝔪$ its maximal ideal, and $L$ an extension of $K$, defined by a monic irreducible polynomial $F\left(X\right)\in R\left[X\right]$. Assume that $\overline{F}\left(X\right)$ factors as a product of $r$ distinct powers of monic irreducible polynomials. In this paper a condition which guarantees the existence of exactly $r$ distinct valuations of $K$ extending $\nu$ is given, in such a way that it generalizes the results given in the paper “Prolongations of valuations to finite...

We consider function field analogues of the conjecture of Győry, Sárközy and Stewart (1996) on the greatest prime divisor of the product $(ab+1)(ac+1)(bc+1)$ for distinct positive integers $a$, $b$ and $c$. In particular, we show that, under some natural conditions on rational functions $F,G,H\in ℂ\left(X\right)$, the number of distinct zeros and poles of the shifted products $FH+1$ and $GH+1$ grows linearly with $degH$ if $degH\ge max\{degF,degG\}$. We also obtain a version of this result for rational functions over a finite field.