On an application of the $abc$-theorem for polynomials to the squarefree neighbour problem
Tom 206 / 2022
Acta Arithmetica 206 (2022), 189-195
MSC: Primary 11C08; Secondary 12D05.
DOI: 10.4064/aa220729-26-10
Opublikowany online: 21 November 2022
Streszczenie
For an integer polynomial $f(x)$, let $L(f)$ denote the sum of the absolute values of its coefficients. Let $N_{f}=d(1+3.6\log (L(f)+1))$ where $\deg f =d$. It is established that there is a positive integer $n\in (N_{f}, 2N_{f}]$ and a $\delta \in \{0,1\}$ such that $x^{n}+f(x)+\delta $ is squarefree. This improves the corresponding bounds obtained by Dubickas and Sha on the degree of the nearest squarefree integer polynomial to $f(x)$. This improvement is a consequence of an application of the $abc$-theorem for polynomials and a lower bound result due to Smyth on the Mahler measure of nonreciprocal integer polynomials.
