On the powerful values of polynomials over number fields

Abstract

Let $\mathcal{B}=\{b_i{\}}_{i=1}^{\infty }$ be a fixed sequence of pairwise distinct elements of a number field $k$. Given the integers $2\le s\le r$, assuming a quantitative version of Vojta's conjecture on the bounded degree algebraic numbers on a number field $k$, we provide lower and upper bounds for the cardinal number of ${\mathbf{G}}_{r,s}^{{\mathcal{B}}_M}$ the set of polynomials $f\in k[x]$ of degree $r\ge 2$ whose irreducible factors have multiplicity strictly less than $s$ and $f(b_1),\cdots ,f(b_M)$ are nonzero $s$-powerful elements in $k$, where $M=2r^2+6r+1$ if $r=s$, and $2s{r}^2+sr+1$ otherwise. Moreover, considering certain conditions on $\mathcal{B}$, we show the existence of an integer $M_0>M$ such that no polynomial in ${\mathbf{G}}_{r,s}^{{\mathcal{B}}_M}$ takes $s$-powerful values at all of $b_1,\cdots ,b_n$ for $n\ge M_0$.

arXiv commentThis is an ubdate version of the paper with modified proof of the main results