Upcoming Seminars for Spring Semester
Unless otherwise noted, the Spring Semester Seminars will be held on the third and fourth Thursday of the month from 1:00pm - 2:00pm in UBC 192.
February 18, 2015; 9:00am in PB 390: Khang Tran (Truman State University)
Title: Zero Distribution of Sequences of Polynomials.
Abstract: We introduce an approach which shows that the zeros of various sequences of polynomials lie on fixed curves on the complex plane. In a simple application of this approach, we prove that the zeros of the polynomial, obtained from the expansion of the binomial polynomial by only keeping the terms whose exponents are in the same congruence class modulus n, lie on the n radial rays emanating from the origin. In another application, we study the complex zeros of a sequence of polynomials satisfying a three-term recurrence of degree n. In particular, given any two polynomials A(z) and B(z) with complex coefficients, we form a sequence of polynomials defined recursively by taking A(z) times the previous polynomial in the sequence plus B(z) times the n-th polynomial before that. We prove that with appropriate initial conditions, the zeros of all the large degree polynomials in this sequence lie exactly on a curve whose equation is given explicitly by the two polynomials A(z) and B(z). The zeros will be dense on this curve when the degree approaches infinity. At the end of the talk, we provide a list of experimental conjectures to which this approach may apply. The author appreciates any possible extension of this list from the audience.
If you need a disability-related accommodation or wheelchair access information, please contact the Mathematics Department at 559.278.2992 or e-mail firstname.lastname@example.org. Requests should be made at least one week in advance of the event.