# Polynomials with symmetric zeros

## TL;DR

This paper reviews polynomials with zeros symmetric to the real line or unit circle, classifying them into self-conjugate, self-inversive, and self-reciprocal types, and discusses their zero distributions.

## Contribution

It provides a concise survey of symmetric-zero polynomials, clarifying their classifications and zero distribution properties.

## Key findings

- Classification of symmetric-zero polynomials into three main types.
- Analysis of zero distribution patterns for each class.
- Highlighting the importance of these polynomials in mathematics and physics.

## 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.