On zeros of self-reciprocal polynomials

2012/11/13 by Masatoshi Suzuki, Suzuki, Masatoshi
Mathematics · #30C15 #34A55 #34L40 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Number Theory (math.NT) #math.CA #math.FA #math.NT #msc:30C15 #msc:34A55 #msc:34L40

paper · pdf · doi:10.48550/arxiv.1211.2953

28 pages; v2 29 pages, revision of Sec.8, typos corrected, refs. added

arxiv created 2012/12/14 · arxiv updated 2012/12/18

Abstract

We establish a necessary and sufficient condition for all zeros of a self-reciprocal polynomial to lie on the unit circle. Moreover, we relate the necessary and sufficient condition with a canonical system of linear differential equations (in the sense of de Branges). This relationship enable us to understand that the property of a self-reciprocal polynomial having only zeros on the unit circle is equivalent to the positive semidefiniteness of Hamiltonians of corresponding canonical systems.

Related

• Unimodularity of zeros of self-inversive polynomials
• The number of unimodular zeros of self-reciprocal polynomials with\n coefficients in a finite set
• Exploring the zeros of real self-reciprocal polynomials by Chebyshev polynomials
• Self-inversive polynomial and quasi-orthogonality on the unit circle
• Improved lower bound for the number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set