Power free values of reducible polynomials

Question

Browning in this paper proves that if $f \in \mathbb{Z}[x]$ is an irreducible polynomial of degree $d \geq 3$ and $k$ is an integer $\geq 3d/4 + 1/4$, then we have $$\#\{n \in \mathbb{Z} \cap [1, X] : f(n) \text{ is $k$-free}\}\sim C_{1}(k, f)X$$ as $X \rightarrow \infty$ where $$C_{1}(k, f) = \prod_{p}\left(1 - \frac{\varrho(p^{k})}{p^{k}}\right)$$ and $\varrho_{f}(n) = \#\{a \pmod{n}: f(a) \equiv 0 \pmod{n}\}$. Is there anywhere that gives a similar asymptotic for the case when $f$ is reducible?

Answer 1

The reducible case is not much harder than the irreducible case, if one uses an argument in Greaves's paper (used originally by Gouvea and Mazur) "Power-free values of binary forms". In particular, the main term can be defined just as easily for reducible polynomials $f(x) \in \mathbb{Z}[x]$. The error terms can be dealt with simply as follows. Write $f(x) = f_1(x) \cdots f_s(x)$, where $f_i(x)$ is irreducible over $\mathbb{Z}$ and define the error term $E_i(X) = \# \{x \in \mathbb{Z} : p^k | f(x) \text{ for some } x > \xi\}$, where $\xi = \frac{1}{k} \log X$. Then the overall error term is just $E(X) = \sum_{j=1}^s E_i(X)$. Thus one can use whatever techniques one would like (in this case, Browning's result is the strongest; although using a different determinant method I am able to reproduce the same result) to estimate each of the $E_i(X)$ and multiply the result by $s$, which is negligible since it is an absolute constant and the error term will be a power saving over the main term (assuming the constant in front of the main term is non-zero).

I should remark that the above estimates do not cover the case when say $f_1(x) = uv^{k-l}$ and $f_2(x) = u'v^l$, or when three factors of $f$ conspire to give us a non-$k$-free term. However, this situation is easily shown to give a negligible contribution. To see this, consider the equations $f_1(x) = uv^{k-l}$, $f_2(x) = u'v^l$ and $f(x) = wv^k$. With $x$ and $v$ fixed, we see that $u, u'$ are divisors of $w$ and hence there are no more than $d(w) = O(w^\epsilon)$ many choices. A simple analysis on the size of $w$ yields that $w = O(X^d)$, and so $d(w) = O(X^\epsilon)$. We may then crudely estimate the number of points lying on the intersection of the varieties defined by $f_1(x) = uv^{k-l}$ and $f_2(x) = u'v^l$ where we take $u, u'$ to be fixed as follows. For $u, u'$ fixed, each of the equations define a curve in $\mathbb{A}^2$, and they have at most finitely many intersection points. Thus the overall contribution is $O(X^\epsilon)$ and is negligible.