ON POWERFUL VALUES OF POLYNOMIALS OVER NUMBER FIELDS

Abstract

Let $\mathcal B =\{b_i\}^{\infty}_{i=1}$ 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 over a number field $k$, we provide lower and upper bounds for the cardinal number of the set $\mathrm G^{\mathcal B_M}_{r,s}$ 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 $2sr^2 + sr + 1$ otherwise. Moreover, considering certain conditions on $\mathcal B$, we show the existence of an integer $M_0 > M$ such that no polynomial in $\mathrm G^{\mathcal B_M}_{r,s}$ takes $s$-powerful values at all $b1,\cdots, b_n$ for $n \geq M_0$.