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Results on Resultants

by

Professor Pedro Teixeira
Union College

January 24, 2005
4:30 pm
Blue House

Tea and pastries will be served


Abstract:

While it's well known that a polynomial equation f(x)=0 with complex coefficients always has a solution (Fundamental Theorem of Algebra), finding the solution(s) is often a very challenging problem. Equations of degree 2 are handled by the quadratic formula, and formulas of a similar flavor (but a lot "messier") take care of equations of degrees 3 and 4. However, for degree 5 and higher no similar formulas exist. In view of this fact, it's surprising that given any two polynomial equations f(x)=0 and g(x)=0, it's always possible to determine whether the equations have a common solution, just using the coefficients of f and g and the arithmetic operations... and it isn't hard either! One possible method is to use the Euclidean Algorithm to find the GCD of f and g. In this talk we'll focus on a second method, which is by using resultants. We'll see how resultants can also be used to determine whether a polynomial has multiple roots, to "solve" parametric equations, and to help solving systems of polynomial equations in two variables, and how, unlike the Euclidean Algorithm, the resultant generalizes in many directions.


For additional information, send e-mail to math@union.edu or call (518) 388-6246.
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