The Brun–Hooley sieve for 𝔽₂[<i>X</i>] and squarefree shifts of integer polynomials

Abstract Let f ( x ) and g ( x ) be polynomials in $\mathbb F_{2}[x]$ with ${\rm deg}\text{ } f=n$ . It is shown that for $n\gg 1$ , there is an $g_{1}(x)\in \mathbb F_{2}[x]$ with ${\rm deg}\text{ } g_{1}\leqslant \max\{{\rm deg}\text{ } g, 6.7\log n\}$ and $g(x)-g_{1}(x)$ having $ \lt 6.7\log n$ terms such that $\gcd(f(x), g_{1}(x))=1$ . As an application, it is established using a result of Dubickas and Sha that given $f(x)\in \mathbb F_{2}[x]$ of degree $n\geqslant 1$ , there is a separable $g(x)\in 2[x]$ with ${\rm deg}\text{ } g= {\rm deg}\text{ } f$ and satisfying that $f(x)-g(x)$ has $\leqslant 6.7\log n$ terms. As a simple consequence, the latter result holds in $\mathbb Z[x]$ after replacing ‘number of terms’ by the L 1 -norm of a polynomial and $6.7\log n$ by $6.8\log n$ . This improves the bound $(\log n)^{\log 4 +\operatorname{\varepsilon}}$ obtained by Filaseta and Moy.