January 31
Speaker: Alexander Borisov (Binghamton)
Title: The abc-polynomials
Abstract: To every triple of coprime natural numbers a=b+c, one can associate an interesting polynomial $f_{abc}(x)=\frac{bx^a-ax^b+c}{(x-1)^2}$. These abc-polynomials were introduced and studied in my old (1998) paper:
http://people.math.binghamton.edu/borisov/documents/papers/abc.pdf In that paper I proved several results on the irreducibility of these polynomials, with the main theorem being that almost all of them, in the density sense, are irreducible. These results can certainly be strengthened and generalized in a number of ways, and the primary reason why this hasn't been done is that, probably, few people tried. I will give an overview of the motivation, results, and methods of this paper, in the hope that some of the graduate students in the audience get interested in pushing this topic further.