Oct 8 2009 - 4:30pm
Oct 8 2009 - 5:30pm
Robert Guralnick
University of Southern California
CH 240
A permutation polynomial over a field F is one that is bijective. If
F is finite and large enough (compared to the degree), then it turns
out that permutation polynomials are bijective on infinitely many extensions
and is called an exceptional polynomial. This goes back to Dickson's thesis
in the 1890's. There is a corresponding notion of maps between smooth curves
(and over arbitrary fields). One of the major conjectures in the field was
the Carlitz conjecture (if F has odd characteristic, then f exceptional implies
it has odd degree). This was solved in 1993 as a part of a much larger program
to classify all exceptional polynomials. A nice mix of group theory and curve
theory is needed to attack this problem. I will discuss the classification
of indecomposable exceptional polynomials whose degree is not a power
of the characteristic.
F is finite and large enough (compared to the degree), then it turns
out that permutation polynomials are bijective on infinitely many extensions
and is called an exceptional polynomial. This goes back to Dickson's thesis
in the 1890's. There is a corresponding notion of maps between smooth curves
(and over arbitrary fields). One of the major conjectures in the field was
the Carlitz conjecture (if F has odd characteristic, then f exceptional implies
it has odd degree). This was solved in 1993 as a part of a much larger program
to classify all exceptional polynomials. A nice mix of group theory and curve
theory is needed to attack this problem. I will discuss the classification
of indecomposable exceptional polynomials whose degree is not a power
of the characteristic.