Discriminants of Polynomials Related to Chebyshev Polynomials: The 'Mutt and Jeff' Syndrome

TL;DR

This paper investigates the factorization of discriminants of polynomials related to Chebyshev polynomials, revealing a surprising similarity in roots and prime factors despite coefficient size differences.

Contribution

It uncovers a novel 'Mutt and Jeff' phenomenon where related polynomials share roots and prime factors in their discriminants, despite coefficient disparities.

Findings

Discriminants factor into two polynomials with dissimilar coefficients.

The polynomials have nearly identical roots.

Discriminants involve the same prime factors.

Abstract

The discriminants of certain polynomials related to Chebyshev polynomials factor into the product of two polynomials, one of which has coefficients that are much larger than the other's. Remarkably, these polynomials of dissimilar size have "almost" the same roots, and their discriminants involve exactly the same prime factors.