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The Brun–Hooley sieve for 𝔽2[X] and squarefree shifts of integer polynomials
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 67 / Issue 4 / November 2024
- Published online by Cambridge University Press:
- 06 November 2024, pp. 1171-1195
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- Article
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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 L1-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.
