Mathematics Colloquium



Department of Mathematics
University of Nebraska at Omaha


WHEN:
Friday February 29, 2008 at 2:30 PM

WHERE:
Durham Science Center, Room 115

WHAT:


Dr. Dave Roberts

University of Minnesota at Morris


will give a talk on

Chebyshev Covers and Exceptional Number Fields


ABSTRACT:
Among the simplest of the classical polynomials are the Chebyshev polynomials of the first and second kind, Tk(x) and Uk(x). In our normalization, the indices are allowed to be half- integers as well as integers, and the "polynomials" actually live in Z[x,(2-x)1/2,(2+x)1/2].   

We will show that the rational functions [(Tm/2 (x)n)/(Tn/2(x)m)] and [(Um/2(x)2n)/(Un/2(x)2m)] are very remarkable from the point of view of Grothendieck's dessins d'enfants. The fibers of these rational functions are likewise very remarkable from the point of view of algebraic number theory. For example, for (m,n) = (125,128) the fiber of the second function above 5 is given by a degree 15875 polynomial in Z[x] with discriminant -2130729 563437 and Galois group the entire symmetric group S15875  





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