arXiv manuscript No. (will be inserted by the editor)
The necessary and sufficient condition for an algebraic integer to be a Salem number
arXiv:1706.01767v1 [math.NT] 6 Jun 2017
Dragan Stankov
Abstract We present a necessary and sufficient condition for a root greater than 1 of a monic reciprocal polynomial of an even degree at least 4, with integer coefficients, to be a Salem number. We also show a method for calculation the minimal polynomial of any power of the root and some properties of its coefficients. Keywords Salem number · Q -linearly independent numbers · reciprocal polynomial · Galois automorphism · Graeffe’s method
1 Introduction A Salem number is an algebraic integer τ > 1 of degree at least 4, conjugate to τ −1 , all of whose conjugates, excluding τ and τ −1 , lie on |z| = 1. The corresponding minimal polynomial P (x) of degree d of these numbers, called a Salem polynomial, is (self-)reciprocal, that is xd P (1/x) = P (x). It is well known [12] that τ n should be also a Salem number of degree d for any natural n. In [14] Vieira, extending a result of Lakatos and Losonczi [6], presented sufficient condition for a monic reciprocal polynomial of an even degree at least 4, with integer coefficients, to be a Salem polynomial. Here we present, in a sense, the opposite direction of this theorem. Namely, we shall prove the following Theorem 1 A real algebraic integer τ > 1 is a Salem number if and only if its minimal polynomial P (x) is reciprocal of even degree d ≥ 4, and there is n ∈ N, n ≥ 2 such that τ n has minimal polynomial Pn (x) = a0,n + a1,n x + · · ·+ ad,n xd D. Stankov Katedra Matematike RGF-a Universiteta u Beogradu, Serbia E-mail:
[email protected]
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Table 1 Salem number τ and n such that the minimal polynomial Pn (x) of τ n satisfy (1) No.
d
Salem number τ
Coefficients
n: Pn (x) satisfy (1)
1. 2. 3. 4. 5. 6. 7.
4 6 8 10 10 10 10
1.722083805739042245 1.506135679553838824 1.280638156267757596 1.216391661138265091 1.230391434407224702 1.261230961137138851 1.176280818259917506
1 1 1 1 1 1 1
9, 13, 16, 17, 20, 24, 27, 31 14, 16, 35, 37, 54, 65, 67 72 53 240 43, 80 >400
-1 -1 -1 0 -1 0 0 -1 -1 0 0 0 -1 -1 0 0 -1 0 -1 0 -1 0 0 -1 1 0 -1 -1 -1
which is also reciprocal of degree d, and satisfies the condition 1 |ad−1,n | > 2
d d−2
(2 +
d−2 X
|ak,n |).
(1)
k=2
We present a method, easy for implementation, for calculation coefficients of Pn (x) starting with the polynomial P (x) without determination of its roots. We can use companion matrix C of a monic polynomial P (x) = xd + ad−1 xd−1 + ad−2 xd−2 + · · · + a0 defined as
0 ··· 0 0 1 ··· 0 0 . . . .. .. .. .. C = . 0 0 0 ··· 0 1 −a0 −a1 −a2 · · · −ad−2 −ad−1 d×d 0 0 .. .
1 0 .. .
It is well known [7], [8] that P (x) is the characteristic polynomial of C so the root λ of P (x) is an eigenvalue of C. If v is an eigenvector of C associated with λ then C n v = C n−1 Cv = C n−1 λv = · · · = λn v. Thus C n should has an eigenvalue λn and the characteristic polynomial of C n must be Pn (x), i.e. Pn (x) = det(xI − C n ). It is easy to show that v = [1 λ λ2 . . . λd−1 ]T . Using this method we are able, for a Salem number τ , to find n such that the minimal polynomial Pn (x) of τ n satisfy condition (1). In the Table 1 we present examples of Salem numbers and n which we have found. The last example in the table is the root of Lehmer polynomial which is the smallest known Salem number. We did not succeeded for it to find n such that Pn (x) satisfy the condition, but we are sure that n > 400. We can notice that n become rarer as d increases. It would be interesting to find n for all small Salem numbers in the Mossinghoff’s list [10]. The last example in the Table 1 shows that, even the degree of P (x) is small, determination of n such that Pn (x) satisfy condition (1) is difficult. But if we observe coefficients of Pn (x) as n increases, we can notice some regularities which enable us to recognize a minimal polynomial of a Salem number. We present these regularities in the Theorem 2.
The condition for an algebraic integer to be a Salem number
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Theorem 2 Let a real algebraic integer τ > 1 be a Salem number and let Pn (x) = 1 + a1,n x + a2,n x2 + · · · + ad−1,n xd−1 + xd be the minimal polynomial of τ n . Then p a n |ad−1,n | = τ , (a) limn→∞ d−1,n+1 ad−1,n = τ , limn→∞ n −n (b) τ + τ ≤ |ad−1,n | + d − 2, (c) for k = 1, 2, . . . , d − 3 d−2 d−2 d−2 |ad−k−1,n | ≤ (|ad−1,n | + d − 2) + + . (2) k k−1 k+1 So if any of the conditions in Theorem 2 is not satisfied we can be sure that a root of P (x) is not a Salem number. Theorem 2 explains the observation that coefficients ak,n for k = 1, 2, . . . , d − 1 of Pn (x) are approximately of the same magnitude, and that a central coefficient is usually slightly greater in modulus than a peripheral one. Examples for this are: P100 , P150 , P200 , showed in Table 2. The alghorithm for calculation a root of P (x) presented in (a) is known as Graeffe’s method [9]. We present one more fact which is useful for recognition of the type of a polynomial P (x), on the basis of its sequence Pn (x). If P (x) is monic, reciprocal, with integer coefficients then Pn (x) is a periodic sequence of polynomials if and only if P (x) is the product of cyclotomic polynomials. Really, if Pn (x) is a periodic sequence, among these polynomials there are only finitely many distinct ones. Then the set of roots of these polynomials is also finite, and all the powers α, α2 , α3 , . . . of a root α of P (x) are in this set. Therefore for some p, q, αp = αq , p 6= q. Since α 6= 0 it follows that αp−q = 1. Vice versa, if P (x) is the product of cyclotomic polynomials then all its roots are roots of 1 so the set of its powers is finite and the set of coefficients ak,n for k = 1, 2, . . . , d − 1, n = 1, 2, . . . of Pn (x) is also finite. Thus Pn (x) is a periodic sequence of polynomials. 2 Proofs of Theorems In order to prove Theorem 1 we shall use a theorem of Kronecker [1, Theorem. 4.6.4.], which is a consequence of Weyl’s theorems. Suppose α = (αk )1≤k≤p ∈ Rp has the property that the real numbers 1, α1 , . . . , αp are Q-linearly independent, and let µ, denote an arbitrary vector in Rp , N an integer and ε a positive real number. Then Kronecker’s theorem states that there exists an integer n > N such that knαk − µk k < ε, (k = 1, . . . , p) where kxk = min{|x − m| : m ∈ Z} is the distance from x to the nearest integer. Proof of Theorem 1 Necessity. Suppose that τ > 1 is a Salem number. The essence of the proof is to show that there is n such that Pn (x) could be in a sense arbitrarily close to Qn (x) = (x − τ n )(x − τ −n )(xd−2 + 1) and that Pn (x) will satisfy the condition (1). It is obvious that roots of (xd−2 + 1) are exp(±( π+2jπ d−2 i), j = 0, 1, . . . , d/2 − 2. Denote conjugates of τ by τ −1 , exp(±2iπω1 ), . . . , exp(±2iπωd/2−1 ).
(3)
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Fig. 1 If τ is the sixth Salem number in Table 1 of degree 10 then conjugates of τ 43 (represented with ◦) and of τ 80 (represented with +), whose minimal polynomials P43 (x), P80 (x) satisfy (1), are close to roots of x8 + 1, vertices of the regular octagon.
It is fulfilled [1, Theorem 5.3.2.] that the numbers 1, ω1 , . . . , ωd/2−1 are Q-linearly independent. According to the Kronecker’s theorem consider 1/2+1 1/2+d/2−2 (wj )2≤j≤d/2 ∈ Rd/2−1 with µ = ( 1/2+0 ). It is clear d−2 , d−2 , . . . , d−2 that for every ε > 0 there exists an arbitrarily large integer m such that |mωj − 1/2+j−1 d−2 | < ε (mod 1) (j = 1, 2, . . . , d/2 − 1). Since a coefficient of a polynomial is a continuous function of its roots, for every ǫ > 0 there exists an arbitrarily large integer n such that the minimal polynomial Pn (x) = Pd/2−2 (x − τ n )(x − τ −n )(xd−2 + 1 + j=1 ǫj (xd−2−j + xj ) + ǫd/2−1 xd/2−1 ), of the Salem number τ n satisfies |ǫk | < ǫ, k = 1, . . . , d/2 − 1. If we denote −τ n − τ −n = T then Pd/2−2 Pn (x) = (x2 + T x + 1)(xd−2 + 1 + j=1 ǫj (xd−2−j + xj ) + ǫd/2−1 xd/2−1 ), Pd/2−1 Pn (x) = xd + 1 + (T + ǫ1 )(xd−1 + x) + (ǫ2 + T ǫ1 + 1)(xd−2 + x2 ) + j=3 (ǫj + T ǫj−1 + ǫj−2 )(xd−j + xj ) + (2ǫd/2−2 + T ǫd/2−1)xd/2 .
The condition for an algebraic integer to be a Salem number
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Table 2 Coeficients of P43 (x), P80 (x) which satisfy (1) and of P100 (x), P150 (x), P200 (x) which do not, where P (x) is the minimal polynomial of the sixth Salem number in Table 1 P43 (x)
P80 (x)
P100 (x)
P150 (x)
P200 (x)
1 -21586 3611 688 5418 -6193
1 -115763027 23986075 -39926871 20167702 4830711
1 -12007769482 29164508197 -18134706516 -25180138718 52256753515
1 -1315811000401282 627267578003835 -2101868014245789 3456484216139549 -1431731747979572
1 -144186527874521531930 415053787386817223949 -542626204385602820124 625113687841885675082 -707660656174865919717
d (2 + It is obvious that |ad−1,n | = |T + ǫ1 | ≥ |T | − |ǫ| and that 12 d−2 Pd−2 k=2 |ak,n |) = Pd/2−1 1 d |ǫj +T ǫj−1 +ǫj−2 |+|2ǫd/2−2 +T ǫd/2−1|) ≤ j=3 2 d−2 (2+2|ǫ2 +T ǫ1 +1|+2 P d/2−1 d 1 j=3 (ǫj + |T |ǫj−1 + ǫj−2 ) + 2ǫd/2−2 + 2 d−2 (2 + 2ǫ2 + 2|T |ǫ1 + 2 + 2 |T |ǫd/2−1 ) ≤ Pd/2−1 d 1 j=3 (ǫ + |T |ǫ + ǫ) + 2ǫ + |T |ǫ) = 2 d−2 (2 + 2ǫ + 2|T |ǫ + 2 + 2 1 d 2 d−2 (4 + 4(d/2 − 2)ǫ + (2(d/2 − 2)) + 1)|T |ǫ) = 1 d 2 d−2 (4 + (2d − 8)ǫ + (d − 3)|T |ǫ). So the condition (1) will be satisfied if d (4 + (2d − 8)ǫ + (d − 3)|T |ǫ) which is equivalent to |T | − |ǫ| > 21 d−2 (2d − 4)|T | − 4d > ǫ, d ≥ 4. (d2 − 3d)|T | + 2d2 − 6d − 4
(4)
Since |T | = τ n + τ −n tends to ∞ as n → ∞ it is obvious that the left side of (2d−4) (4) tends to D := (d 2 −3d) as n → ∞. Thus if we choose ǫ such that D > ǫ > 0 then there is an integer N such that (4) will be fulfilled for all n ≥ N . Since n can be arbitrarily large there is n ≥ N such that Pn (x) satisfies |ǫk | < ǫ, k = 1, . . . , d/2 − 1 which is sufficient, as we showed, to be proved that Pn (x) satisfies (1). Sufficiency. Suppose that τ > 1 is a real algebraic integer with conjugates τ1 = τ , τ2 , . . . , τd over Q. If τ is conjugate to τ ′ then τ n is conjugate to τ ′n . Since minimal polynomial Pn (x) of τ n is of degree d so τ1n , τ2n , . . . , τdn must be different numbers and their product have to be 1 because Pn (x) is monic and reciprocal. From condition (1) by applying the main theorem in [14] it follows that there are d − 2 of them on the boundary of the unit disc |z| = 1. Since they occur in conjugate complex pairs their product is equal to 1. It follows that τ −n should be a conjugate of τ n which allow us to conclude that τ n is a Salem number. If |τ ′n | = 1 then |τ ′ | = 1 thus it follows that there are d − 2 conjugates of τ on the boundary of the unit disc. Finally, in the same manner as for τ n , we conclude that τ is a Salem number.
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Proof of Theorem 2 (a) Since τ n is a Salem number Pn (x) has to be monic, reciprocal polynomial of even degree. Using the notation (3) for conjugates of τ and Vieta’s formulae we have Pd/2−1 τ n+1 + τ −n−1 + k=1 e2(n+1)iπωk + e−2(n+1)iπωk ad−1,n+1 = . Pd/2−1 ad−1,n τ n + τ −n + k=1 (e2niπωk + e−2niπωk )
If we divide the enumerator and the denominator with τ n then it is obvious that we obtain the enumerator which tend to τ and the denominator which tend to 1, as n → ∞. If we write d/2−1
|ad−1,n | = τ n |1 + τ −2n + τ −n
X
k=1
e2niπωk + e−2niπωk |,
p then we can conclude immediately that limn→∞ n |ad−1,n | = τ . Pd/2−1 (b) Since ad−1,n = −τ n − τ −n − k=1 e2niπωk + e−2niπωk it is obvious that τ n + τ −n ≤ |ad−1,n | + d − 2. (c) From (3) we have d/2−1
Pn (x) = (x − τ n )(x − τ −n )
Y
x − e2niπωk
k=1
Pn (x) = (x2 − (τ n + τ −n )x + 1)
d−2 Y
m=1
x − e−2niπωk
(−1)m 2niπω m ⌈2⌉ x−e
If we denote the product by Qn (x) = 1+b1,nx+b2,n x2 +· · ·+bd−3,n xd−3 +xd−2 then we can see that bk,n , k = 1, 2, . . . , d − 3 is the sum of d−2 summands k where each of them is of modulus 1 so that |bk,n | ≤ d−2 . We can conclude k that (2) is valid by expanding Pn (x) = (x2 − (τ n + τ −n )x + 1)Qn (x) and using (b).
References 1. Bertin, M.-J., Decomps-Guilloux, A., Grandet-Hugot, M., Pathiaux-Delefosse, M., and Schreiber, J.-P.: Pisot and Salem numbers, Birkh¨ auser, Basel, (1992) 2. Boyd, D.W.: Pisot and Salem numbers in intervals of the real line. Math. Comput. 32, 1244-1260 (1978) 3. Boyd, D.W.: Small Salem numbers. Duke Math. J. 44, 315-328 (1977) 4. Ghate, E.; Hironaka, E., The arithmetic and geometry of Salem numbers. Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 3, 293-314. 5. Lakatos, P.: On zeros of reciprocal polynomials. Publ. Math. Debr. 61, 645-661 (2002) 6. Lakatos, P., Losonczi, L.: Self-inversive polynomials whose zeros are on the unit circle. Publ. Math. Debr. 65, 409-420 (2004) 7. P. Lancaster and M. Tismenesky, The Theory of Matrices, Academic, (1985) 8. R. Lidl and H. Niederreiter, Introduction to Finite Fields” and Their Applications, Cambridge U.P., (1986)
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9. J. M. McNamee and V. Pan, Numerical Methods for Roots of Polynomials, Part II, vol. 16 of Studies in Computational Mathematics, Elsevier Science, Cambridge, Mass, USA, (2013) 10. Mossinghoff M., List of small Salem numbers, http://www.cecm.sfu.ca/~mjm/Lehmer/lists/SalemList.html 11. Schinzel, A.: Self-inversive polynomials with all zeros on the unit circle. Ramanujan J. 9(12), 19-23 (2005) 12. Smyth, C.: Seventy years of Salem numbers: a survey. Bull. Lond. Math. Soc. 47(3), 379-395 (2015) 13. Stankov, D.: On the distribution modulo 1 of the sum of powers of a Salem number. Comptes Rendus - Mathematique, 354(6), 569–576 (2016) 14. Vieira, R.S.: On the number of roots of self-inversive polynomials on the complex unit circle. Ramanujan J. 42(2), 363–369 (2017)
