Abstract

Polynomials whose zeros are symmetric either to the real line or to the unit circle are very important in mathematics and physics. We can classify them into three main classes: the self-conjugate polynomials, whose zeros are symmetric to the real line; the self-inversive polynomials, whose zeros are symmetric to the unit circle; and the self-reciprocal polynomials, whose zeros are symmetric by an inversion with respect to the unit circle followed by a reflection in the real line. Real self-reciprocal polynomials are simultaneously self-conjugate and self-inversive so that their zeros are symmetric to both the real line and the unit circle. In this survey, we present a short review of these polynomials, focusing on the distribution of their zeros.

Highlights

  • IntroductionWe consider the theory of self-conjugate (SC), self-reciprocal (SR), and self-inversive (SI) polynomials

  • The basic properties of these polynomials can be found in the books of Marden [1], Milovanović et al [2], and Sheil-Small [3]. These polynomials are very important in both mathematics and physics, it seems that there is no specific review about them; in this work, we present a bird’s eye view to this theory, focusing on the zeros of such polynomials

  • The theorems below are usually applied to positive self-reciprocal (PSR) polynomials, but some of them can be extended to negative self-reciprocal (NSR) polynomials as well

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Summary

Introduction

We consider the theory of self-conjugate (SC), self-reciprocal (SR), and self-inversive (SI) polynomials. ZsðzÞ 1⁄4 ðzs þ zÀsÞ for any integer s can be written as a polynomial of degree s in the new variable x 1⁄4 z þ 1=z (the proof follows by induction over s); we can write pðzÞ 1⁄4 zmqðxÞ, where qðxÞ 1⁄4 q0 þ ⋯ þ qmxm is such that the coefficients q0, ..., qm are certain functions of p0, ..., pm. From this we can state the following: Theorem 3.

How these polynomials are related to each other?
Zeros location theorems
Polynomials that do not necessarily have symmetric zeros
Real self-reciprocal polynomials
Complex self-reciprocal and self-inversive polynomials
Section 5.1
Where these polynomials are found?
Polynomials with small Mahler measure
Knot theory
Bethe equations
À 3t þ 3t2 À 3t3 þ t4 1 À 3t þ 5t2 À 3t3 þ t4
Orthogonal polynomials
Conclusions
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