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 $2\leq s \leq r$, 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 $r\geq 2$ 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 $2sr^2+ s r+1$ 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 $n\geq M_0$.