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  • J. W. BOBER (a1) (a2), D. FRETWELL (a3) (a4), G. MARTIN (a5) and T. D. WOOLEY (a6)


Given $f\in \mathbb{Z}[t]$ of positive degree, we investigate the existence of auxiliary polynomials $g\in \mathbb{Z}[t]$ for which $f(g(t))$ factors as a product of polynomials of small relative degree. One consequence of this work shows that for any quadratic polynomial $f\in \mathbb{Z}[t]$ and any $\unicode[STIX]{x1D700}>0$ , there are infinitely many $n\in \mathbb{N}$ for which the largest prime factor of $f(n)$ is no larger than $n^{\unicode[STIX]{x1D700}}$ .


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The third author’s work is partially supported by a National Sciences and Engineering Research Council of Canada Discovery Grant. The fourth author’s work is supported by a European Research Council Advanced Grant under the European Union’s Horizon 2020 research and innovation programme via grant agreement no. 695223.



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  • J. W. BOBER (a1) (a2), D. FRETWELL (a3) (a4), G. MARTIN (a5) and T. D. WOOLEY (a6)


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