Results on Resultants
Professor Pedro Teixeira
January 24, 2005
Tea and pastries will be served
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.
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