How to count the number of zeros that a polynomial has on the unit circle?

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:384

Article

Vieira, R. S.¹ 

[1] Univ Fed Sao Carlos UFSCar, Dept Matemat, Rod Washington Luis,Km 235 SP-310, BR-13565905 Sao Carlos, SP, Brazil

关键词: Self-inversive polynomials;    Self-reciprocal polynomials;    Salem polynomials;    Sturm theorem;    Mobius transformations;    Cayley transformations;   

DOI  :  10.1016/j.cam.2020.113169

来源: Elsevier

【 摘 要 】

The classical problem of counting the number of real zeros of a real polynomial was solved a long time ago by Sturm. The analogous problem of counting the number of zeros that a polynomial has on the unit circle is, however, still an open problem. In this paper, we show that the second problem can be reduced to the first one through the use of a suitable pair of Mobius transformations - often called Cayley transformations - that have the property of mapping the unit circle onto the real line and vice versa. Although the method applies to arbitrary complex polynomials, we discuss in detail several classes of polynomials with symmetric zeros as, for instance, the cases of self-conjugate, self-adjoint, self-inversive, self-reciprocal and skew-reciprocal polynomials. Finally, an application of this method to Salem polynomials and to polynomials with small Mahler measure is also discussed. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

Free