An inequality on the greatest roots of a polynomial (Q1186827)

Summary: Let \(\rho\) be the greatest modulus of the roots of a monic polynomial \(P\) with complex coefficients and height \(H\). We prove \(\rho<1+H^{1/k}\) if \(P\) has \(k\) roots of modulus \(\rho\) (for \(k=1\), this is due to Cauchy). In particular, when \(P\) is a polynomial with real coefficients which has a non real root \(\alpha\) then \(|\alpha|<1+\sqrt H\).