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Abstract:
An amoeba is the set of all points (log|z_1|, ... log|z_n|), where
(z_1, ... ,z_n) are the solutions to a system of polynomial equations.
The name comes from the fact that pictures of these sets have a
tendency
to look like single celled organisms.
Amoebas haved proved to be important in a number of different
parts of math, including a recent partial solution to a Hilbert
problem.
Yet the most basic question, "given a point in R^n, how can one tell
if it's in the amoeba" is non-trivial. The simplest case of this problem
is: given a single polynomial f(z) in one variable, determine whether
it has a root on the unit circle. I will talk about an approach to
this
problem, based on the triangle inequality, which turns out to
generalise
nicely to the case of systems of polynomials in several variables.
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