Towards Turán's polynomial conjecture

Abstract

We revisit an old problem posed by P. Turán asking whether there exists an absolute constant $C>0$ such that if $f\left(x\right)\in Z\left[x\right]$ with $degf=d$, then there is a polynomial $w\left(x\right)\in Z\left[x\right]$ with $degw⩽d$ and the sum of the absolute values of the coefficients of $w\left(x\right)$ is less than or equal to $C$ such that the sum $f\left(x\right)+w\left(x\right)$ is irreducible over $Q$. In 1970, A. Schinzel obtained a partial answer establishing the existence of an $w\left(x\right)$ for $C=3$ with ${degw<\exp \left((5d+7)\right(\parallel f\parallel }^2+3\left)\right)$ where ${\parallel f\parallel }^2$ is the sum of the squares of the coefficients of $f\left(x\right)$. Schinzel's result was further refined by P. Banerjee and M. Filaseta in 2010, obtaining $degw\ll d\exp (8\parallel f{\parallel }^2+9)$ with an absolute implied constant. The main contribution of the current work lies in establishing the existence of $w\left(x\right)$ for $C=6$ with a significantly improved upper bound on $degw$. For instance, our main result implies the existence of an explicit $w\left(x\right)$ for $C=6$ with $degw\ll d{\log }^2(\parallel f\parallel +2)$, where the implied constant is absolute.