cv.complex variables - Minimizing the modulus of a polynomial around a circle

There is another way.
Every non-negative trigonometric polynomial $f$ on the circle
is of the form $|q|^2$, where $q$ is an analytic polynomial.
(I mean by this that $f$ is of form $sum_{-N}^N a_n z^n$ and
$q(z) = sum_0^N b_n z^n$).
This is called the Fejer-Riesz theorem.

So, you guess a minimum for $|p|^2$, call it $m$, and then see
whether $f = |p|^2 - m$ is the modulus squared of a polynomial
(an algebraic identity).
If it is, try again with larger $m$; if not, reduce $m$.

For a fuller account, see the survey article by Helton and Putinar:

@incollection {MR2389626,
AUTHOR = {Helton, J. William and Putinar, Mihai},
TITLE = {Positive polynomials in scalar and matrix variables, the
spectral theorem, and optimization},
BOOKTITLE = {Operator theory, structured matrices, and dilations},
SERIES = {Theta Ser. Adv. Math.},
VOLUME = {7},
PAGES = {229--306},
PUBLISHER = {Theta, Bucharest},
YEAR = {2007},
MRCLASS = {47-02 (14P10 47A13 47A57 47A63 90C22)},
MRNUMBER = {MR2389626 (2009i:47001)},
MRREVIEWER = {Joseph A. Ball},
}

-John E. McCarthy