Re: a polynomial with positive real roots

From: Amanda (sca18_at_hotmail.com)
Date: 01/11/05 

Date: 11 Jan 2005 04:21:21 -0800

David C. Ullrich wrote:
> On 10 Jan 2005 12:00:38 -0800, "Amanda" <sca18@hotmail.com> wrote:
>
> >
> >david petry wrote:
> >> Amanda wrote:
> >>
> >> > Dealing with an algorithm, I came across the following problem:
> >> >
> >> > I have a polynomial P of degree>1 and real coefficients. I know
> >that,
> >> > if z is a root of P, then Re(z)>=0. It follows that, if r is a
real
> >> > root of P', then r>=0.
> >> >
> >> > I can affirm this because all roots of P lie in the closed half
> >plane
> >> > to the right of the imaginary axis. According to Lucas' theorem,
it
> >> > follows all the roots of P' lie in this same half plane, which
> >> implies
> >> > that, if r is a real root of P', than r>=0.
> >> > This conclusion is correct, isn't it?
> >>
> >> The conclusion is correct, though I'm not sure what Lucas' Theorem
> >> is. The roots of P' lie in the convex hull of the roots of P.
> >
> >Lucas' theorem says that, if all roots of P are in a same open half
> >plane of the complex plane, then the roots of P' are in this same
half
> >plane. I think Lucas' theorem implies the roots of P' lie in the
convex
> >hull of the roots of P.
>
> Yes, that's equivalent, because a convex set is an intersection
> of half-planes.
>

Lucas's theorem deals only with open half planes, right? That is, it's
proof assumes the roots of P are in an open half plane, isn't it?
Amanda