Annular bounds for the zeros of a polynomial from companion matrices

摘要

Let p(z) = z(n) + a(n-1)z(n-1) + a(n-2)z(n-2) + ... + a(1)z + a(0) be a complex polynomial with a(0) not equal 0 and n >= 3. Several new upper bounds for the moduli of the zeros of p are developed. In particular, if alpha = root Sigma(n-1)(j=0) vertical bar a(j)vertical bar(2) and z is any zero of p, then we vertical bar z vertical bar(2) <= cos(2) pi/n+1 + vertical bar a(n-2)vertical bar + 1/4 (vertical bar a(n-1)vertical bar + alpha)(2) + 1/2 root alpha(2) - vertical bar a(n-1)vertical bar(2) + 1/2 alpha, which is sharper than the existing bound, given as, vertical bar z vertical bar(2) <= cos(2) pi/n+1 + 1/4 (vertical bar a(n-1)vertical bar + alpha)(2) + alpha, if and only if 2 vertical bar a(n-2)vertical bar < root Sigma(n-1)(j=0) vertical bar a(j)vertical bar(2) - root Sigma(n-2)(j=0) vertical bar a(j)vertical bar(2): The upper bounds obtained here enable us to describe smaller annuli in the complex plane containing all the zeros of p.