On the properties of fibotomic polynomials

TL;DR

This paper investigates the algebraic properties of fibotomic polynomials, including their discriminants, resultants, factorizations over prime fields, and generalizations to bivariate cases, advancing understanding of their structure.

Contribution

It introduces the concept of fibotomic polynomials and provides comprehensive properties, including discriminants, resultants, and factorizations, extending to bivariate forms.

Findings

Discriminant formulas for fibotomic polynomials

Resultants of pairs of fibotomic polynomials

Complete factorization over prime fields

Abstract

Define the $n$-th fibotomic polynomial to be the product of the monic irredicible factors of the $n$-th Fibonacci polynomial which are not factors of any Fibonacci polynomial of smaller degree. In this paper, we prove a number of properties of the fibotomic polynomials. This includes determining the discriminant of the fibotomic polynomials and the resultant of pairs of fibotomic polynomials. Furthermore, we completely determine the factorization form of the fibotomic polynomials in prime fields. Results are also generalized for the bivariate homogenous fibotomic polynomials.