Is the constant in the abc-conjecture dependent on $\epsilon$ only?

Question

Consider the ABC conjecture in the following form:

For every positive real number $ϵ$, there exists a constant $k_\epsilon$ such that for all triples $(a,b,c)$ of coprime positive integers, with $a+b=c$:

$c<k_\epsilon⋅rad(abc)^{1+ϵ}$

Is the constant in the abc-conjecture dependent on $\epsilon$ only?

For example, if I chose

$k_\epsilon=1/c^\epsilon$

or some other function of $c$ and $\epsilon$, is the $c$ dependence allowed in the constant or is it an $\epsilon$ only choice?

Answer 1

The constant depends only on $\epsilon$. This is implied by the structure of the statement: the quantifier "there exists a constant $k_\epsilon$..." comes before "for all triples $(a,b,c)$..." so the constant cannot depend on the triple (since the statement must hold for all triples after the constant has been chosen). The notation $k_\epsilon$ also is a reminder that it should depend only on $\epsilon$ (similar notations are commonly used to indicate which parameters a constant can depend on).

Note that if $k_\epsilon$ was allowed to depend on $c$ then the statement would be trivial since there are only finitely many choices of $a$ and $b$ such that $a+b=c$ so you can just pick $k_\epsilon$ big enough to work for all of them.