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                Department of Mathematical Sciences
                   The Johns Hopkins University

                             SEMINAR

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Steven Zelditch                                        February 21, 2002
Department of Mathematics                              304 Whitehall Hall 
The Johns Hopkins University                           Refreshments: 3:30 PM 
                                                       Seminar: 4:00 PM 

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                    STATISTICAL PATTERNS IN ZEROS

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                             ABSTRACT

It has been known for many years that the Newton polytope of a polynomial 
in several variables has a strong impact on its zeros.  (The Newton 
polytope is simply the convex hull of the exponents of the terms 
occuring in the polynomial).  For instance, Kouchnirenko, Bernstein 
and others in the 70's proved that the number of simultaneous zeros 
of a system  k polynomials in k (complex) variables with Newton polytope P 
equals k! Vol(P).   My talk is about the impact of P on the distribution 
of zeros.  In recent work with Bernard Shiffman, we have proved that the 
zeros of k polynomials in k variables with Newton polytope P tend to
concentrate in P, and only a very small number manage to `tunnel' out.
As I will explain, this is a result on random polynomials with a fixed
Newton polytope and of high degree.  I hope to make the ideas accessible 
to a general audience.