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The Polynomial Pell Equation and its Analogs (March 2013)
On behalf of the Department of Mathematics and Statistics, AUS College of Arts and Sciences, you are cordially invited on a seminar to be conducted by Dr. Timothy Marshall, Associate Professor of Mathematics and Statistics, AUS.
Abstract
One of the best known Diophantine equations is Pell's equation x^2 - dy^2 = 1, where d is a square free integer, and x and y are required to be integers. The solutions of Pell's equation form a group, and in fact this is well known to be a cyclic. We can also consider polynomial versions of this equation. The best known example is the equation P^2(x) + (1 - x^2)Q^2(x) = 1, Where P(x) and Q(x) are now required to be polynomials in C[x]. The solutions are well known: (P(x), Q(x)) = ( +(-) T_n(x) , + (-) U_{n - 1}(x)), where n in Z, and T_n(x) and U_{n - 1}(x) are the Chebyshev polynomials of the first and second kind. We outline a topological proof of this (under the additional assumption that the coefficients in P and Q are real). If time permits, I will also talk vaguely about 4-term analog of this equation.
For further details, kindly contact [email protected].