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SMPT Lecture Series: Lecture No. 1 for 042015-012016

posted Sep 1, 2015, 11:54 AM by smpt cob   [ updated Sep 1, 2015, 12:02 PM ]

School of Mathematics, Physics and Technology
September 8th
Room T1, 2 p.m.
At how many points do two curves intersect?
Dr. Pinaki Mondal
School of Mathematics, Physics and Technology
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 n-dimensional 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) Bernstein-Kushnirenko 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 Bernstein-Kushnirenko result.