The Resultant of Chebyshev Polynomials

Abstract

Abstract. Let Tn denote the n-th Chebyshev polynomial of the first kind, and let Un denote the n-th Chebyshev polynomial of the second kind. We give an explicit formula for the resultant res (Tm, Tn). Similarly, we give a formula for res(Um, Un).

The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials

“…The first formula for the resultant of two cyclotomic polynomials was given by Apostol [3]. Some other papers have been dedicated to the study of resultant of Chebyshev Polynomials [6,19,24,28]. In this paper we deduce simple closed formulas for the resultants of a big family of GFPs.…”

Resultants of Chebyshev Polynomials: A Short Proof

“…(e.g., see [JRT,(3.3)]). To begin with, res(T m , T n ) = 0 if and only if there exist k and l such that…”

“…Proposition 1 (See [JRT,Corollary 4.2]) For m, n ≥ 1, we have res(T m , T n ) = 0 if and only if m 1 := m/ gcd(m, n) and n 1 := n/ gcd(m, n) are odd.…”

“…Theorem 3 (See [JRT,Theorem 4.5]) For m, n ≥ 1 we have res(T m , T n ) = 0 m/ gcd(m, n) and n/ gcd(m, n) are odd, 2 mn−m−n+gcd (m,n) otherwise.…”

“…If g = 1, then m 1 = m + 1 must divide k < m and n 1 = n + 1 must divide l < n, which cannot occur. Hence res(T m , T n ) = 0 and res(U m , U n ) = 0 if and only if gcd(m + 1, n + 1) > 1, as in [JRT,Corollary 5.2]. So we now assume that m 1 := m + 1 and n 1 := n + 1 are coprime.…”