Resultants of Chebyshev Polynomials: the First, Second, Third, and Fourth Kinds

Home , Cyclotomic polynomial

Canad.Math. Bull. Vol. þÇ (z), z§Ëþ pp. zh– hË http://dx.doi.org/Ë§. Ëþh/CMB-z§Ëþ-§§z-Ç ©Canadian Mathematical Society z§Ëþ

Masakazu Yamagishi

Abstract. We give an explicit formula for the resultant of Chebyshev polynomials of the ûrst, second, third, and fourth kinds. We also compute the resultant of modiûed cyclotomic polynomials.

1 Introduction In [ ], Jacobs, Rayes, and Trevisan obtained explicit formulas for the resultants of Chebyshev polynomials of the ûrst and second kinds, and Louboutin gave a short proof in [Ç]. As there are four (ûrst, second, third, and fourth) kinds of Chebyshev polynomials, it is the purpose of this note to compute the resultant of two Chebyshev polynomials of any kinds.It is intriguing to notice that the Jacobi symbol is involved in the result. For the proof, we use the roots of Chebyshev polynomials, basic properties of sine and cosine values, and basic properties of the Jacobi symbol, including the reciprocity law. When restricted to the ûrst or second kinds, our proof is diòerent from both [ ] and [Ç]. As an application, we also compute the resultant of modiûed cyclotomic polynomials.Our result is a reûnement of a well-known formula due to Diederichsen [z] (see also [Ë, h,þ, 6,]) for the resultants of cyclotomic polynomials.

2 Resultant of Chebyshev Polynomials _e Chebyshev polynomials , , , and of the ûrst, second, third, and fourth Tn Un Vn Wn kind, respectively, are characterized by

sin(n + Ë)θ T (cos θ) = cos nθ, U (cos θ) = , n n sin θ cos(n + Ë~z)θ sin(n + Ë~z)θ V (cos θ) = , W (cos θ) = , n cos θ~z n sin θ~z where n is an integer (cf. [ËË,Ëz]). _e normalized Chebyshev polynomials of the ûrst and second kinds are deûned by z z , z . We adopt Cn(x) = Tn(x~ ) Sn(x) = Un(x~ ) Schur’s notation . For odd we deûne Sn(x) = Sn−Ë(x) n z , z . Vn(x) = V(n−Ë)~z(x~ ) Wn(x) = W(n−Ë)~z(x~ )

Received by the editors October h, z§Ëh. Published electronically March z, z§Ëþ. _is work was supported by JSPS KAKENHI (No. zhþ §§Ë ). AMS subject classiûcation: ËËR§, ËËRËÇ, ËzEË§, hhC þ. Keywords: resultant,Chebyshev polynomial, cyclotomic polynomial. zh

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In this note, we always assume odd indices for the polynomials and . For Ë Vn Wn n ≥ we have

z cos jπ , z cos jπ , Cn(x) = ∏ x − Sn(x) = ∏ x − §< j

(z.Ë) Ë n , Wn(−x) = (− ) Vn(x) ⎧ ⎪W (x)~W (x) if n is odd, (z.z) W (C (x)) = ⎨ mn n m n ⎪ ( )~ ( ) if is even. ⎩⎪Smn~z x Sn~z x n Let res( f , g) denote the resultant of two polynomials f and g. For the deûnition and properties of the resultant, see [ ]. In particluar, we note that

(z.h) res(g, f ) = (−Ë)deg( f ) deg(g) res( f , g). In addition to those properties listed in [ ], we also quote the following from [Ë§].If h is a polynomial with leading coeõcient c, then (z. ) res f (h(x) , g(h(x)) = cdeg( f ) deg(g) deg(h) res( f , g)deg(h).

Let = r ei be the prime factorization of a positive odd integer , and an n ∏i=Ë p n a integer. _e Jacobii symbol is deûned by

a r a ei (z.þ) = ∏ , n i=Ë pi where is the Legendre symbol. For a prime and an integer we write (a~pi ) p n ord when k is the highest power of dividing . p(n) = k p p n

_eorem z.Ë Let m, n be positive integers and let g = gcd(m, n). ⎧ ⎪(−Ë)mn~zzg if ord (m)= ~ ord (n), (i) res(C , C ) = ⎨ z z m n ⎪§ ord ( ) = ord ( ). ⎩⎪ if z m z n ⎧ ⎪(−Ë)(m−Ë)(n−Ë)~z if g = Ë, (ii) res(S , S ) = ⎨ m n ⎪§ > Ë. ⎩⎪ if g (iii) res( , ) = m . Vm Vn n (iv) res , n (Wm Wn) = . ⎧ m ⎪(−Ë)m(n−Ë)~zzg−Ë if ord (m) ≥ ord (n), (v) res(C , S ) = ⎨ z z m n ⎪§ ord ( ) < ord ( ). ⎩⎪ if z m z n (vi) res(C , ) = res( , C ) = z z(g−Ë)~z. m Vn Wn m n~g (vii) res( , ) = res( , ) = m . Sm Vn Wn Sm n (viii) res( , ) = z z(g−Ë)~z. Wm Vn g

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Proof For positive integers n and a we introduce the following products: ajπ ajπ S(n, a) = ∏ z sin , C(n, a) = ∏ z cos , §< j

res , z cos jπ z cos njπ (Cm Cn) = ∏ Cn = ∏ §< j

Remark z.z In [ ,Ç], e xplicit formulas for res , and res , were ob- (Tm Tn) (Um Un) tained. _ey are identical to our (i) and (ii), respectively, in view of (z. ).

Lemma z.h Let n, a be positive integers and g = gcd(n, a). ⎧ ⎪(−Ë)(n−Ë)(a−Ë)~zn if g = Ë, (i) ( , ) = ⎨ S n a ⎪§ ⎩⎪ otherwise.

Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 03:11:39, subject to the Cambridge Core terms of use. z@ M. Yamagishi ⎧ ⎪(−Ë)(n−Ë)a~zzg−Ë if ord (n) ≤ ord (a), (ii) ( , ) = ⎨ z z C n a ⎪§ ⎩⎪ otherwise. (iii) If n is odd, then ⎧ √ ⎪ a n if a is odd, , n s(n a) = ⎨ ~ √ ⎪ a z n if a is even. ⎩ n (iv) If n is odd, then ⎧ ⎪ z z(g−Ë)~z if a is odd, c(n, a) = ⎨ g ⎪ z z(g−Ë)~z if a is even. ⎩ n~g

Proof We put n′ = n~g, a′ = a~g. (i)I f Ë, then , §, since sin ′ §. Suppose Ë and let g > S(n a) = (an π~n) = g = ζN = zπi~N ( is a positive integer). Since z sin( ~ ) = −a j(Ë − a j), we have e N ajπ n iζzn ζn

n−Ë ( , ) = n−Ë −an(n−Ë)~z Ë − j = (−Ë)(n−Ë)(a−Ë)~z . S n a i ζzn ∏ ζn n j=Ë (ii)I f ord ord , then , §, since cos ′ z §. Suppose z(n) > z(a) C(n a) = (a(n ~ )π~n) = ord ord ; in particular, ′ is odd.If Ë, then by (i) we have , z(n) ≤ z(a) n g = S(n a)= ~ §, S(n, za)= ~ §, and

S(n, za) C(n, a) = = (−Ë)(n−Ë)a~z. S(n, a)

If g > Ë, then

′ g−Ë n −Ë a(kn′ + j)π g−Ë a(kn′)π C(n, a) = ∏ ∏ z cos × ∏ z cos k=§ j=Ë n k=Ë n − ′ ′ − g Ë n −Ë ′ a jπ g Ë ′ = (−Ë)a k z cos × z(−Ë)a k ∏ ∏ ′ ∏ k=§ j=Ë n k=Ë ′ = (−Ë)(n−n )a~zzg−ËC(n′, a′)g ,

and we are reduced to the case g = Ë. (iii)I f > z, then ( , ) = § since sin( ′ ~ ) = §. Suppose = Ë. It follows from g S n a an π n g √ (i) and the identity S(n, a) = (−Ë)(n−Ë)(a−Ë)~zs(n, a)z that Ss(n, a)S = n. By ~ Gauss’ Lemma, the sign of s(a, n) is equal to a z if a is even (cf. [@, Propo- n sition Ç.Ë]). If a is odd, then, counting the number of odd j’s in the interval § < j < n~z, we see that s(n, a) = z s(n, a + n), so the sign of s(n, a) is a . n n (iv)I f g = Ë, then by (iii) we have s(n, a)= ~ § and ⎧ s(n, za) ⎪Ë if a is odd, c(n, a) = = ⎨ s(n, a) ⎪ z if a is even. ⎩ n

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Suppose g > Ë. We compute (g−Ë)~z a(kn′ + j)π c(n, a) = ∏ ∏ z cos k=§ §< j

z where ε = Ë if a′ is even, and ε = (−Ë)(g −Ë)~Ç = z if a′ is odd. _us we are g reduced to the case g = Ë.

3 Resultant of Modiﬁed Cyclotomic Polynomials For h let Ψ denote the minimal polynomial of z cos z over . _en Ψ n ≥ n ( π~n) Q n(x) ∈ and deg Ψ z. _ese are what we called the modiûed cyclotomic poly- Z[x] ( n) = ϕ(n)~ nomials in the introduction, as we have the identity

(h.Ë) Ψ −Ë −ϕ(n)~zΦ , n(x + x ) = x n(x) where Φ is the -th cyclotomic polynomial. Here are some properties of Ψ . For n n n n odd we have (h.z)Ψ Ψ . zn(x) = n(−x) By [Ëh, Proposition z.þ] we have

⎧ ( ~ ) ⎪∏ W (x)µ n d if n ≡ Ë (mod z), ⎪ dSn d µ(n~zd) (h.h)Ψ (x) = ⎨∏ S ~ V (x) if n ≡ z (mod ), n ⎪ d n z d ⎪ µ(n~zd) if § mod . ⎩∏dSn~z Sd (x) n ≡ ( ) Combining this with (z.z), for p a prime we have ⎧ ⎪Ψ (x)Ψ (x) if p ∤ n, (h. ) Ψ (C (x)) = ⎨ pn n n p ⎪Ψ ( ) if S . ⎩⎪ pn x p n We need to generalize the Jacobi symbol to the Kronecker symbol; it is deûned by (z.þ) for any positive integer n by requiring further that ⎧ ⎪§ if a ≡ § (mod z), a ⎪ = ⎨Ë if ≡ ±Ë (mod Ç), z a ⎪ ⎩⎪−Ë if a ≡ ±h (mod Ç). We introduce one more notation. For a positive integer n let L(n) = p if n is a power of some prime p, and L(n) = Ë otherwise. _e notation Λ(n) = log L(n) is

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oen used in analytic number theory. We note the following identity:

(h.þ) L(n) = ∏ d µ(n~d). dSn

Lemma h.Ë Let n ≥ h. (i)Ψ z n( ) = L(n). (ii) Ψ z z Except for (− ) = − , we have ⎧ − ⎪ Ë if n is odd, Ψ z L(n) n(− ) = ⎨ ⎪ −Ë L(n~z) if n is even, n > . ⎩ L(n~z) (iii) Ψ § §, Ψ § z Except for ( ) = Ç( ) = − , we have ⎧ −z if n ≡ Ë (mod z), ⎪ L(n) ⎪ Ψ (§) = ⎨ z if n ≡ z (mod ), n L(n~z) ⎪ ⎪ −Ë L(n~ ) if n ≡ § (mod ). ⎩ L(n~ )

_e exceptional cases are clear as Ψ , Ψ z z. For odd the Proof (x) = x Ç(x) = x − n claims follow from (h.h), (h.þ), and the facts (z) = n, (−z) = −Ë , (§) = Wn Wn n Wn −z . _en applying (h. ) for p = z (so C (x) = xz − z), we complete the proof. n z Now we compute res Ψ , Ψ . In view of (z.h), we may make some additional ( m n) restrictions on m and n.

_eorem h.z Let m, n ≥ h, m =~ n. (i) If m ∤ n, n ∤ m, and m is odd, then L(n) res(Ψ , Ψ ) = . m n L(m) (ii) If m ∤ n, n ∤ m, m < n, and m, n are even, then L(m~z) res(Ψ , Ψ ) = . m n L(n~z) (iii) , If m S n and m is odd, then, putting LË = L(m) Lz = L(n~m), we have ⎧ res Ψ , Ψ ⎪ Lz , ( m n) ⎪ if LË =~ Lz = ⎨ LË ϕ(m)~z ⎪Ë if L = L . Lz ⎩ Ë z (iv) ord Ë z , If m S n and z(m) = , then, putting LË = L(m~ ) Lz = L(n~m), we have ⎧ res Ψ , Ψ ⎪ Lz , ( m n) ⎪ if LË =~ Lz = ⎨ LË ϕ(m)~z ⎪ −Ë . ⎪ if LË = Lz Lz ⎩ LË (v) If S m S n, then, putting Lz = L(n~m), we have ⎪⎧−Ë if m = , n = Ç, res(Ψ , Ψ ) ⎪ m n = ⎨ −Ë = , =~ Ç, ( )~ if m n Lϕ m z ⎪ Lz z ⎩⎪Ë otherwise.

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Proof (i) We use induction on the number of prime divisors (multiplicity taken into account) of g = gcd(m, n). If g = Ë, then the claim follows from (h.h), _eorem z.Ë, and (h.þ); note that L(n) = Ë if n ≡ z (mod ) and that L(n~z) = L(n) if n ≡ § (mod ). Suppose g > Ë. Since the assumption implies L(n) = Ë or L(m) = Ë, we have to show that res Ψ , Ψ Ë. Let be a prime divisor of . We have h, z ( m n) = p g m~p ≥ n~p ≥ by the assumption. If n~p = z, then by (z.h), (z. ), and (h.z) we have res Ψ , Ψ res Ψ , Ψ res Ψ , Ψ , ( m n) = zm(−x) p(−x) = ( p zm) whose computation will be postponed until (iii). Suppose n~p ≥ h. By (z. ) we have res Ψ , Ψ p res Ψ , Ψ , ( m~p n~p) = ( m~p(Cp(x)) n~p(Cp(x)) then by (h. ) we ûnd that ⎧ res Ψ , Ψ p−Ë ⎪ ( m~p n~p) ⎪ , if µ = ν = Ë, ⎪res(Ψ , Ψ ) res(Ψ , Ψ ) ⎪ m n~p m~p n ⎪res Ψ , Ψ p ⎪ ( m~p n~p) ⎪ , if µ = Ë, ν ≥ z, (h.@) res(Ψ , Ψ ) = ⎨ res(Ψ , Ψ ) m n ⎪ m~p n ⎪res Ψ , Ψ p ⎪ ( m~p n~p) ⎪ , if µ ≥ z, ν = Ë, ⎪ res(Ψ , Ψ ~ ) ⎪ m n p ⎪res Ψ , Ψ p, if z, z, ⎩⎪ ( m~p n~p) µ ≥ ν ≥ where we put ord , ord . _e idea of using this identity is borrowed µ = p(m) ν = p(n) from [h]. In each case, it follows from the induction hypothesis that res Ψ , Ψ Ë. ( m n) = (ii) _e case is immediate from Lemma h.Ë, since res Ψ , Ψ Ψ § . We m = ( n) = n( ) suppose that m ≥ @ and use the identity (h.@), which is valid also for p = z. ord ord Ë By (z.h),(h.@), and (i) we have _e case z(m) = z(n) = . L(n~z) res(Ψ , Ψ ) = (−Ë)ϕ(m~z)ϕ(n~z)~ . m n L(m~z) Since we have ϕ(k) ≡ L(k) − Ë (mod ), if k ≥ h is odd, we have L(m~z) res(Ψ , Ψ ) = m n L(n~z) by the reciprocity law. ord Ë, ord z Similarly, we have _e case z(m) = z(n) ≥ . L(n) res(Ψ , Ψ ) = . m n L(m~z) We have either L(n) = L(n~z) = Ë or L(n) = L(n~z) = z, so, in any case, L(n) L(m~z) = . L(m~z) L(n~z) ord z, ord Ë Similarly, we have _e case z(m) ≥ z(n) = . L(m) res(Ψ , Ψ ) = (−Ë)ϕ(m)ϕ(n~z)~ . m n L(n~z)

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Since we have S ϕ(m), z S ϕ(n~z), and L(m) = L(m~z), we are done. ord z, ord z Similarly, we have res Ψ , Ψ Ë. Since at least _e case z(m) ≥ z(n) ≥ . ( m n) = one of m~z, n~z is not a prime power, we are done. (iii) We use induction on the number of prime divisors (multiplicity taken into account) of m.Let p be a prime divisor of m. _e case m = p. If n = zp, then by _eorem z.Ë we have z res(Ψ , Ψ ) = res(W , V ) = z(p−Ë)~z, m n p p p as desired. Suppose n~p ≥ h. First we compute (p−Ë)~z z res Ψ , Ψ Ψ z cos jπ Ψ z (p−Ë)~z, ( p n~p(Cp)) = ∏ n~p Cp = n~p( ) j=Ë p so that by Lemma h.Ë we have (h.6) res Ψ , Ψ ϕ(p)~z. p n~p(Cp) = L(n~p) Now, if ord Ë, then we have p(n) =

res(Ψ , Ψ ~ (C )) L(n~p) res(Ψ , Ψ ) = p n p p = ( ~ )ϕ(p)~z p n res Ψ , Ψ L n p ( p n~p) L(p) by (i) and (h.6). If ord z, then we have p(n) ≥ res Ψ , Ψ res Ψ , Ψ ϕ(p)~z. ( p n) = p n~p(Cp) = L(n~p) Since either L(n~m) = Ë or L(n~m) = L(m) = p holds, we are done. ord Ë, In this case Ë, so we have to show that _e case p(m) = m > p. L(m) = res Ψ , Ψ ϕ(m)~z. First suppose ord Ë. By (h.@) and the induc- ( m n) = L(n~m) p(n) = tion hypothesis we have res Ψ , Ψ ϕ(m~p)~z, res Ψ , Ψ Ë. ( m~p n~p) = ±L(n~m) ( m~p n) = Since , , and Ë, we have res Ψ , Ψ Ë by (i). _us m ∤ (n~p) (n~p) ∤ m L(m) = ( m n~p) = we have res Ψ , Ψ ϕ(m)~z. We next suppose ord z, and use (h.@) ( m n) = L(n~m) p(n) ≥ and the induction hypothesis.If Ë, then Ë, so res Ψ , Ψ Ë. Lz = L(n~(m~p)) = ( m n) = Otherwise, , so Lz = L(n~(m~p)) = p p p−Ë res(Ψ , Ψ ) = pϕ(m~p)~z = pϕ(m)~z. m n L(m~p) ord z By equation (h.@) and the induction hypothesis, and noting _e case p(m) ≥ . that L(m~p) = L(m), we obtain the desired result. (iv) ord Ë _e case z(n) = . By (z. ) and (h.z) we have res Ψ , Ψ Ë ϕ(m)ϕ(n)~ res Ψ , Ψ . ( m n) = (− ) ( m~z n~z) We could use (h.@) for p = z to deduce this. As is easily seen, we have ⎧ ⎪Ë if L =~ L , (−Ë)ϕ(m)ϕ(n)~ = ⎨ Ë z ⎪ −Ë if L = L , ⎩ LË Ë z

Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 03:11:39, subject to the Cambridge Core terms of use. Resultants of Chebyshev Polynomials hË so the claim follows from (iii). ord z We use equation (h.@) for z and (iii). If Ë, then _e case z(n) ≥ . p = Lz = z Ë, so res Ψ , Ψ Ë. Otherwise, z z and , L(n~(m~ )) = ( m n) = Lz = L(n~(m~ )) = LË =~ Lz so z res Ψ , Ψ zϕ(m~z)~z. ( m n) = LË (v) _is is immediate from Lemma h.Ëi f m = . Otherwise, using (h.@) for p = z and induction, we complete the proof.

Corollary h.h ([Ë–h,þ, 6,]) If h ≤ m < n, then ⎧ ⎪Ë if m ∤ n, res(Φ , Φ ) = ⎨ m n ⎪ ( ~ )ϕ(m) S . ⎩⎪L n m if m n

Proof Let ζ = ezπi~m. By (h.Ë) we have res Ψ , Ψ z Ψ j − j j −ϕ(n)~zΦ j res Φ , Φ , ( m n) = ∏ n(ζ + ζ ) = ∏ (ζ ) n(ζ ) = ( m n) × × j∈(Z~m) j∈(Z~m)

since ∑ × ≡ § (mod ). So the claim follows from _eorem h.z. j∈(Z~m) j m

References

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