Heights of roots of polynomials with odd coefficients
For an integer \(m\geq 2\) let \(D_m\) denote the set of integer polynomials whose coefficients are all \(\equiv 1\pmod m\). The authors prove: If \(\alpha\) is a zero of a polynomial \(f\) in \(D_m\) of degree \(n\) which is not a \( 2(n+1)\) root of unity (not a \((n+1)\) root of unity if \(m\geq 3\)), then the height of \(\alpha\) satisfies \[ h(\alpha)\geq \frac{c_m}{n+1} \] with \[ c_2=0.427800 \] and \[ c_m=\log\left(\frac m2\right)+\frac{2.947486-\frac {\delta}{2}}{m^2}+O\left(\frac{1}{ m^4}\right), \] where \(\delta=1\) if \(m\geq 3\) odd, \(\delta=0\) if \(m\geq 4\) even. The optimal \(c_m\) satisfies \(c_m=\log m+O(1)\).
scientific article; zbMATH DE number 5862105