On the $k$-free values of the polynomial $xy^k+C$

Kostadinka Lapkova

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arXiv · 1506.08044

On the $k$-free values of the polynomial $xy^k+C$

Kostadinka Lapkova

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Abstract

Consider the polynomial $f(x,y)=xy^k+C$ for $k\geq 2$ and any nonzero integer constant $C$. We derive an asymptotic formula for the $k$-free values of $f(x,y)$ when $x, y\leq H$. We also prove a similar result for the $k$-free values of $f(p,q)$ when $p,q\leq H$ are primes. The strongest tool we use is a recent generalization of the determinant method due to Reuss.

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Kostadinka Lapkova. 2015-06-26. On the $k$-free values of the polynomial $xy^k+C$. https://arxiv.org/abs/1506.08044

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