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Báo cáo toán học: "From a Polynomial Riemann Hypothesis to Alternating Sign Matrices"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: From a Polynomial Riemann Hypothesis to Alternating Sign Matrices. | From a Polynomial Riemann Hypothesis to Alternating Sign Matrices Omer Egecioglu Department of Computer Science University of California Santa Barbara CA 93106 omer@cs.ucsb.edu Timothy Redmond Network Associates Inc. 3965 Freedom Circle Santa Clara CA 95054 redmond@best.com Charles Ryavec College of Creative Studies University of California Santa Barbara CA 93106 ryavec@math.ucsb.edu Submitted March 27 2001 Accepted October 24 2001. MR Subject Classifications 05E35 11M26 12D10 Abstract This paper begins with a brief discussion of a class of polynomial Riemann hypotheses which leads to the consideration of sequences of orthogonal polynomials and 3-term recursions. The discussion further leads to higher order polynomial recursions including 4-term recursions where orthogonality is lost. Nevertheless we show that classical results on the nature of zeros of real orthogonal polynomials i. e. that the zeros of pn are real and those of pn 1 interleave those of pn may be extended to polynomial sequences satisfying certain 4-term recursions. We identify specific polynomial sequences satisfying higher order recursions that should also satisfy this classical result. As with the 3-term recursions the 4-term recursions give rise naturally to a linear functional. In the case of 3-term recursions the zeros fall nicely into place when it is known that the functional is positive but in the case of our 4-term recursions we show that the functional can be positive even when there are non-real zeros among some of the polynomials. It is interesting however that for our 4-term recursions positivity is guaranteed when a certain real parameter C satisfies C 3 and this is exactly the condition of our result that guarantees the zeros have the aforementioned interleaving property. We conjecture the condition C 3 is also necessary. Next we used a classical determinant criterion to find exactly when the associated linear functional is positive and we found that the Hankel determinants An .