LECTURE SERIES SCHOOL OF MATHEMATICS, PHYSICS AND TECHNOLOGY September 8^{th}
Room T1, 2 p.m.
At how many points
do two curves intersect?
Dr. Pinaki Mondal
School of Mathematics,
Physics and Technology
Abstract
Understanding
the behaviour of shapes from their equation is a basic quest in
mathematics. The topic
of the talk will be one aspect of this problem, namely: how to
determine/estimate the number of points of intersection of two curves in
the plane (or more generally, of n hypersurfaces in the ndimensional
space) defined by polynomial equations. The most basic
answer to this question is known as "Bezout's Theorem" (discovered by
Newton about 100 years earlier than Bezout) which says that a sharp
upper bound is given by the product of the degrees of these polynomials.
We will also talk about the mysterious (and, I
am sure you will agree, beautiful) BernsteinKushnirenko theorem, which
gives a better upper bound in terms of volumes of some polytopes
determined by these polynomials. If time permits we will talk about my
work generalizing the BernsteinKushnirenko result.

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