Re: Putnam 2005 -- some answers [SPOILER ALERT]

In article <dmuc6u$sqq$1@xxxxxxxxxxxxxxxxxx>,
David Rusin <rusin@xxxxxxxxxx> wrote:


>A-3.
>
>No idea.

Is this what Rolle's theorem looks like when wrapped around the
unit circle?

A polynomial like p, all of whose roots lie on the unit circle,
is (the numerator of) a composite f( (1-ti)/(1+ti) ), with f a
(multiple of a) real polynomial and t ranging over the real line.
(This is just the Moebius transfomation taking the real line to the
unit circle.) This gives one real root of f' between consecutive
pairs of roots on the real line, so that p' must have a root
on the unit circle between any two roots of p.

But I can't seem to turn this into the desired equation.
Maybe it's just getting late.

dave


.

Relevant Pages

  • Re: An idea for rough estimation of roots of a polynomial
    ... You are right, it must be R>=0,where R < 1 are for zeros or roots ... inside or on the unit circle. ... I have a rough idea in mind for finding the roots of a polynomial. ... the zeropadded vector. ...
    (comp.dsp)
  • Re: Roots of a cubic
    ... in the unit circle without computing them? ... one of the roots is real and is definitely greater than 1. ... The other two are complex conjugates of each ... then the other two roots lie on the unit circle. ...
    (sci.math.research)
  • Re: x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + K*x^6 + x^5 + x^4 + x^3 + x^2 + x +
    ... Show that all roots of flie on the unit circle ... Obviously a similar approach applies to all symmetric polynomials ... require knowing when all roots of a polynomial lie inside the unit ...
    (sci.math)
  • Re: function
    ... You then need to show that the roots of this polynomial lie on the unit circle. ...
    (sci.math.num-analysis)
  • Re: calculating polynomial coefficients from roots
    ... the roots lie on the unit circle in the ... I would like to be able to compute the coefficients for up ... to/from the Schelkunoff polynomial/root representation. ...
    (sci.math.num-analysis)
Quantcast