My question concerns the sums of two reducible polynomials with bounded coefficients. In particular, for every $d \geq 2$ does there exist a number $C(d) > 0$ such that for any two co-prime reducible polynomials $f,g \in \mathbb{Z}[x]$ of degree $d$ there exist integers $u,v$ such that $|u|, |v| \leq C(d)$ and $uf(x) + vg(x)$ is irreducible?
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asked Nov 17, 2016 at 22:15
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$\begingroup$ I'm not sure what you mean by "non-proportional", but if $f(x)=x(x-1)$ and $g(x)=x(x-2)$.... $\endgroup$– Gerry MyersonCommented Nov 17, 2016 at 22:22
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$\begingroup$ Presumably you mean coprime, not non-proportional. Regardless, surely $C$ must be allowed to grow with the degree. For example, choose $f$, $g$, and $uf+vg$ for small $u$ and $v$ to vanish at distinct integers. $\endgroup$– rloCommented Nov 17, 2016 at 22:27
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$\begingroup$ Ah yes, $C$ should be a function of the degree, I will fix that. And yes, co-prime is the appropriate condition $\endgroup$– Stanley Yao XiaoCommented Nov 17, 2016 at 23:00
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1$\begingroup$ I'm not sure C exists for d=0. Do you know of a bound that permits ua+vb to be prime when a and b are composite and coprime integers? Gerhard "Thinking Of Really Simple Cases" Paseman, 2016.11.17. $\endgroup$– Gerhard PasemanCommented Nov 17, 2016 at 23:25
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$\begingroup$ @Gerhard Paseman I put in the condition $d \geq 2$ $\endgroup$– Stanley Yao XiaoCommented Nov 18, 2016 at 6:52
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