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Q,Q2, ' " If Ave the Conjugates of 0 , Then ©I,***,© Are All Root8 of Unity. N Pjz) = = Zn + Bn 1 Zn~L

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Q,Q2, ' ALGEBRAIC INTEGERS ON THE UNIT CIRCLE* J. Hunter (received 15 April, 1981) 1. Introduction An algebraic integer 0 is a complex number which is a root of an irreducible (over Q ) monic polynomial P(s)=an + a^ ^ zn * + • • • + ao , where the a^ are integers. If 0j = Q,Q2 , ' " ,6^ are the roots of P(a) , then ©l.*’**© are called the conjugates of 0 . The first important result connecting algebraic integers and the unit circle was due to Kronecker [3]: Theorem 1 (Kronecker, 1857). If |0^| 5 1 (1 < i < n) , where ©1 * ©»©2 »’**»©„ave the conjugates of 0 , then ©i,***,© are all root8 of unity. We include a proof of Theorem 1 partly for completeness and partly because it contains ideas (especially those connected with the introduc­ tion of polynomials related to P(a) , the monic irreducible polynomial for 0) that are used to prove other results. Let n P J z ) = (m = 1,2,3,...) = zn + bn_ 1 zn ~ l* ••• b+ 0 , say. * This survey article comprises the text of a seminar presented at the University of Auckland on 15 April 1981. Math. Chronicle 11(1982) Part 2 37-47. 37 Then Pj (a) = P(a) . Also, i£ = 0* + ••• + 0*(k i JV) , then £ 77j ( k = 1,2,3, •••) , by Newton's formulae involving symmetric functions of • Since the b^ are symmetric polynomials in 07.,**»©m with integer coefficients, it follows that the b. 1 tt are rational integers. Now |bn = |the sum of the ^.J products of taken i at a time| < since |0™| <1 (1 < r < n) . Since this holds for each m = 1,2,3,••• , it follows that there is only a finite number of distinct polynomials among the Pm (3) • Hence there is an infinite set of indices, {n>i ,^2 ,^3, * • •}, such that P (2) = P (a) = P (a) = ••♦ . Then 0mi,0m2,•••,0m«+l , for each w j m 2 r r r r - !,••*,« , are all roots of the same polynomial of degree n . Since there are n+1 such numbers it follows that 0m£ = 0m«7 k\ r r (m. f m.) and so 0 = 1 for some k ( It. From this the required i r result follows. 2. Related conjectures For P(a) as described in Section 1, let tt E(P) = n m a x (|0 j.l) , £= 1 so that E{P) = 1 when J0^|<1 (1 < t < rt) and E(P) > 1 when |0^| > 1 for at least one i , and then E(P) is the product of those 10^| which are > 1 (i.e. are in "excess" of 1). Theorem 1 can be restated as: "E{P) = 1 =» all the 0^ are roots of unity". The first recorded conjecture on E(P) is due to Lehmer [4], 38 Conjecture 1 (Lehmer, 1933). There is a constant h > 0 , independent of n , such that E(P) < l+h =» 0i»***,0w all roots of unity. Note. If £*^(3) = zn -2 , then its roots lie on the circle \z\ = . Sin.ce 2* n ->1 as «-*■“ , the roots of P (z ) can be taken arbi- n trarily close to |2 | = 1 by taking n large enough, but the roots are not on |3| = 1 Now ^(P^) = 2 for each n and so, in Conjecture 1, we require 0 < h < 1 . Now let M(P) = max |6.| . t A conjecture on M(P) , which is related to Conjecture 1, is due to Schinzel and Zassenhaus [5]: Conjecture 2 (Schinzel and Zassenhaus, 1965). There is a constant c > 0 , independent of n , such that Q M(P) < ! + — => all roots of unity.• From Theorem 1, "W(P) < 1 =» all roots of unity". 3. Recent results Although both of the conjectures (1 and 2) are still unsettled considerable progress has been made in the last ten years (see D.W. Boyd [l]). As with many problems of a number-theoretic nature it is useful to partition the polynomials into two classes. Progress was made when the polynomials P(a) were divided into non-reciprocal and reciprocal polynomials. The best result for the non-reciprocal case is due to Smyth [6]. P(s) is called reciprocal if P(s) = zn > otherwise P(z) is non-reciprocal. If 0 is a root of unity (0^1) , then its monic irreducible polynomial is reciprocal, but the converse statement is not true. 39 Theorem 2 (Smyth, 1971). If P(z) is a non-reciprocal polynomial, then E(P ) > 0O and W(P) > 1 + i log 0O , where 0o = 1.3247... is the real root of the polynomial p 0 (3) * S 3-3-l . This can be stated as: E(P) 00 =» P(a) reciprocal, W(P) < 1 + — log 0o =» P(a) reciprocal. Note. Since E{P0) = 0O , Smyth's result for E(P) is the best possible. The strongest result involving roots of unity that has so far been proved is due to Dobrowolski [2]: Theorem 3 (Dobrowolski, 1981). For all integers n > 2 , (i) E{P) < 1 + (*°|0^ n] - 0i»,,*»8ri oil roots of unity. (ii) W(P) < 1 + l~f~0gnffn] * n“ 9 l’*“ »0n al1 rootB °f unity • 4. A weak form of Smyth's theorem In this partial survey of the topic it may ue of interest to include proofs of weaker forms of Theorems 2 and 3, partly for the results themselves, but mainly for the nature of the proofs. In this section we present a weak form of Smyth's result. Theorem 4. In the usual notation, E(P) < %(1 + /17) =* P(a) reciprocal, or, equivalently, 40 P(s) non-reciprocal =» £’(f>) > % (1 + /IT) [Note that %(1 + /T7) = 1.2807... , as compared with 0 q = 1.3247. in Theorem 2.] For the proof we require a lemma. Lemma. If f(z) = £ e zr (e € JR, e 0 f 0) ia holomorphio in a r=0 r r region containing the diec |a| < 1 and |/(z)| <1 on |a| = 1 , then |e j < 1 - el {k = 1,2,•••) . Proof. Let k be a positive integer and let t be any real number. Then , k - 1 oo / ( a ) ( t + 3 ) = [ (ter )zr + I {ter*er A z V » r=0 r=k and hence, using Parseval's identity, (te0)2 + (tex)2 + •• +te (kl)2 + l ^ tev *er. ^ 2 * \ l/(et8) l2|t+elfc0|2<ie Thus f2" t2e 2 + (fce^+«o)2 - Jir J (£2+2fccosfc0+l)<20 = fc2+l Take t = — sqn e , : then e 0 y /c h i * 1 , e 0 - 2 e o and so (e20 ♦ \ek I)2 < 1 , giving e2 + |e,| < 1 , and hence |e, | < 1 - e 2 (k > 1) 41 Proof of Theorem 4. We consider the second form of the statement. Suppose that P(s) = zn + an _y * + * * ’ + a0 non-reciprocal, so that, if Q(z) = 3nP(Vj,) , then P(z) / Q(z) . If |ero| - 2 , then F(P) > 2 and so we can assume that |P(0)j = |a0| = 1 . P(z) and Q(z) have no common roots. For, if they have a common root, then, since P(s) is irreducible, P(z)|<?(2) . But P(z ) is monic and so Q(0) = 1 . It would then follow that P(z ) = ±Q(z) . P(*0 = Q(z) is impossible and P(z) = -Q(z) implies that -P(1)=<?(1) , whereas Q(z) = znP(V3) gives <2(1) = P(l) ; then P(l) = 0 , contra­ dicting the irreducibility of P(z) . A particular consequence of this discussion on common roots is that 10 -1 / I (1 5 i < n) ; otherwise, on noting that 0. = — follows from _ t 1 = 1 and that 0^ , 0^ are both roots of the real polynomial P , it would follow that P(z) and #(z) would have a common root. Consider now the power series T(P) = P(Q)PW - i + t ,* + ... 1 J Q(z ) 1 'ft* (1) where ft is the smallest exponent with corresponding coefficient + 0 (ft > 1) . We note that i 77, . For, P(z) = zn * an jzn * + ••• a\z+ + a$ , P(0) = clq = +1 or -1 and Q(z) = 3”p( ^ ) = 1 + + ••• clq+ Z*1 , and so T m = P( 0 ) P <3) = ■♦a<K«w. 1»W ~1^ o » W 1+ (°n_ * ’ *a QZn ) = (l+a0ai3+*•+a0an ^zn ^+a0zn)(1-X+X2-X3+ ' •) , 42 where X = + ... + a ^z . Consequently the coefficients in the power series expansion for T{z) are integers. The constant term in (1) is 1 since T(0) = -- = y = 1 . Now, since P(z) ( iF?[a] , n n - n n fl - ) n P{z) = fl (2-9.) = n (2-0.) and Q{z) = zn fl - - 0. = n (1-0.2) . ■ t lz il . ' x i= 1 i= 1 1=1 v ’ 7=1 Hence 2-0 . X p(0) n (2-e.) p(o) n 0 . <1 1-0 .2 i=i * 1 x' X T (2) = = £iil , say. 9(z) 1-0 .2 n (1-0.2) I t n i=i• * 2-0 . Since T(0) = 0 , it follows that /(0) = <7(0) . Hence, by considering -/(a) , if necessary, we can assume that /(0) = #(0) > 0 . Then -g(z) 2-0 . X /(3) = ±p(0) n = C q + C\Z + •• • |e.|<: 1-0 .2 X and 1-0 .2 X n = J n + d\Z + 2-0 | e. | > l 1 x' where O q = d o > 0 . All poles of f(z) and g(z) are in the region |2 | > 1 and f(z) , g{z) are both holomorphic in a region containing the disc \z\ < 1 . 43 Since |e^| < 1 if and only if |0^| < 1 , the non-real 0^ in /(z) and <7 (2 ) give rise to factors of the form 3 - 0 .
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