Staircase palindromic polynomials

TL;DR

This paper introduces staircase palindromic polynomials, a special class of monic-palindromic polynomials, and demonstrates their factorization into cyclotomic polynomials, revealing new algebraic structures and relationships.

Contribution

The paper defines staircase palindromic polynomials and proves their factorization into cyclotomic polynomials, providing a systematic way to derive these factors for any degree n.

Findings

Staircase polynomials can be factored into cyclotomic polynomials.

Number of staircase polynomials for a given degree n is approximately half of n.

Certain classes of polynomials can be transformed into staircase polynomials.

Abstract

We study a class of monic-palindromic polynomials that we call staircase palindromic polynomials. Specifically, suppose $S(x, n, h)$ is a polynomial of degree n with the special form: $S(x; n; h) = x^n + 2x^{n-1} + 3x^{n-2} + \dots + (h - 1)x^{n-h+2} + hx^{n-h+1} + \dots + hx^{h-1} + (h - 1)x^{h-2} + \dots + 2x + 1$. Then $S(x, n, h)$ can be factored as a product of cyclotomic polynomials. Moreover, for any given n, there are $ \lceil{\frac{n+1}{2}\rceil}$ staircase polynomials, all of whose factors can be derived using two parameter n and h with the help of cyclotomic polynomials. After that we explore some classes of polynomials that can be converted to staircase polynomials.