# Discriminants of a Class of Self-inversive Polynomials and Real Binary Forms

## TL;DR

This paper explores the relationships between certain classes of self-inversive polynomials and real binary forms, providing explicit discriminant formulas via determinants of coefficient-dependent matrices.

## Contribution

It establishes bijections between classes of self-inversive polynomials and real binary forms, and derives explicit discriminant expressions using polynomial coefficient matrices.

## Key findings

- Bijections between self-inversive polynomials and real binary forms
- Discriminants expressed as determinants of matrices
- Matrices have entries as polynomials in polynomial coefficients

## Abstract

A class of self-inversive polynomials includes all the self-reciprocal polynomials. Let A denote the set of all self-reciprocal polynomials with n+1 coefficients. Let B denote the set of certain self-inversive and non self-reciprocal polynomials with n+1 coefficients for odd n. Let C denote the set of real binary n-ic forms. Then there exist a bijection between A and C and another bijection between B and C. Let f be a monic polynomial in A and g be the corresponding polynomial in C. If the reading coefficient of g is not zero, then the discriminant of g is expressed by the determinant of a matrix of type (n, n). Any element of the matrix is a polynomial in the coefficients of f with integer coefficients. The same holds for monic polynomials in B.