MATHEMATICS COLLOQUIUM

Symbolic Resultants via Dixon Formulation,

and Their Applications to Geometry

Dr. Arthur Chtcherba

Department of Computer Science

University of Texas-Pan American

As the computing power of modern computers grows, resultants are being used more to solve problems from a variety of areas like computer vision, solid modeling, geometric theorem proving, and robotics. It is also being used to solve numeric problems as the methods based on resultant matrices are much more numerically stable.

Due to its generality, in the past resultants were impractical requiring much computing power. Nowadays problems of practical interest are within reach, and as a consequence the last 12 years have produced more material about resultants, than for the entire preceding 200 year history of the subject.

In this talk, I will present yet another method to compute resultants. So far the method has shown to be most effective theoretically, as well as, in practice, where polynomial systems have ad hoc structures. Based on quite successful Dixon formulation, the method adapts to input and gives smallest answers, implying that the answer has smallest amount of extraneous information.

Resultants can be used to solve problems from geometric reasoning. An usual approach is to describe a geometric setting using polynomial equations, then negate a statement which one wishes to prove and show that the system is inconsistent. Here a different approach is taken. Since the resultant is a natural consequence of the polynomial equations, and hence they are ”theorems”. To prove or discover a geometric theorem one only needs to properly setup a hypothesis and compute the resultant which is a conclusion. We will demonstrate this on the Pappus hexagon theorem and show how one can obtain its generalization, the Pascals ”mystic” hexagon.

Date: Monday, November 15

Time: 3:15pm

Place: J. Wiener Lecture Hall