On the roots of polynomials with concentration at low degrees (Q750659)

The author considers complex polynomials \(\sum^{n}_{j=0}a_ jz^ j\) which have concentration d at degree k, i.e. satisfy \[ \sum^{k}_{j=0}| a_ j| \geq d\sum^{n}_{j=0}| a_ j|,\text{ where } 0<d\leq 1. \] By induction on k the author proves the following main Theorem. For each fixed \(0<d<1\) and natural k there exists a largest open disc with center 0 and radius \(R=R(d,k)\), in which every polynomial with concentration d at degree k has at most k zeros. Furthermore, \(dk^ k/2(k+1)^{k+1}\leq R(d,)\leq 1\) holds. By considering the special class of polynomials with positive coefficients having all zeros in Re \(z\leq 0\) (called Hurwitz polynomials by the author) it is shown that also \(R(d,k)\leq \min \{1,(1-d)^{- 1/(k+1)}-1\}\) holds. Altogether this yields \[ d/2e(k+1)\leq R(d,k)\leq (- \log (1-d))/(k+1). \]