Algebra & Combinatorics Research Group  |  Texas A&M University





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A&C Seminar:  Fall 2008, Fridays, Milner 317, 3:00–3:50 p.m.

September 12 Chris Hillar (TAMU)
3:00–3:50 Polynomial recurrences and cyclic resultants
Abstract:   Given a monic univariate polynomial f(x) of degree d, the m-th cyclic resultant of f is r_m = Resultant(f,x^m-1). These sequences of numbers give coarse information about certain dynamcal systems (toral endomorphims), but they also arise in many other contexts such as number theory, Lagrangian mechanics, varieties of amoebas, and quantum computation. It is an important open question how many such resultants (the coarse data) determine the given polynomial f (the dynamical system). Sturmfels and Zworski have conjectured that d+1 resultants suffice, however, the best current bound is 2^{d+1}. We discuss the most recent attacks on this problem. In the process we will explain how binomial factorizations in group algebras, polynomial recurrences, and Toepletz determinant factorizations play surprising special roles. (partly joint with Lionel Levine, MIT).