ACTA ARITHMETICA XCIII.2 (2000) The Fekete Szegő theorem with slitting conditions: Part I by Robert Rumely (Athens, GA) A classical theorem of Fekete and Szegő [4] says that if E is a comact set in the comlex lane, stable under comlex conjugation and having logarithmic caacity γ(e) 1, then every neighborhood of E contains infinitely many conjugate sets of algebraic integers. Rahael Robinson [5] refined this, showing that if E is contained in the real line, then every neighborhood of E contains infinitely many conjugate sets of totally real algebraic integers. In [2], David Cantor develoed a theory of caacity for adelic sets in P 1. One of his key results was a very strong theorem of Fekete Szegő Robinson tye, which roduced algebraic numbers whose conjugates lay in a secified neighborhood of an adelic set E = E E, and belonged to P 1 (R), and P 1 (Q ) for finitely many rimes ( slitting conditions ). Unfortunately there was a ga in the art of the roof concerning the slitting conditions. Some time ago the author extended Cantor s theory, including the Fekete Szegő theorem without slitting conditions, to arbitrary algebraic curves [6]. This aer reresents a ste towards establishing the theorem with slitting conditions. We rove the theorem in the secial case where the ground field is Q, the sets are E = [ 2r, 2r] and E = Z for rimes in a finite set T, and caacities are measured relative to the oint. It will be aarent to anyone familiar with this kind of result that we have drawn ideas from earlier aers. The method of roof, called atching, goes back to Fekete and Szegő [4]. The use of Chebyshev olynomials for the archimedean atching functions comes from Robinson [5], and the use of Stirling olynomials for the -adic atching functions comes from Cantor [2]. However, we have introduced several new ideas: in articular, the method for reserving well-distributed sequences of roots of -adic olynomials, and 2000 Mathematics Subject Classification: Primary 11R06, 31C15; Secondary 11R04, 14G40. Key words and hrases: Fekete Szegő theorem, caacity theory, slitting conditions. Work suorted in art by NSF grant DMS-9500842. [99]
100 R. Rumely the ste of moving the roots of the artially atched -adic olynomials to kee them well-searated, are new. 1. Statement of the theorem Notations. Q and R are the fields of rational and real numbers, Q is the field of -adic numbers, Z is the ring of -adic integers. The canonically normalized absolute value on Q will be written x, and its associated valuation, ord (x). For the archimedean rime =, we will write Q = R, and x for the usual absolute value x on R. For finite rimes, if x = (x 1,..., x m ) Q m, let ord (x) = min(ord (x i )). Given a set E Q, and a function f : E Q, we write f E for the su norm of f on E with resect to x. We will often measure the distance between -adic numbers α, β in terms of ord (α β): if ord (α β) t, then we say α and β are searated in ord value by at least t; and if ord (α β) t, then α and β are searated in ord value by at most t (note the reversal of comaratives). For a finite set T, #(T ) will denote the cardinality of T. The natural logarithm of a real number t will be written ln(t), and the base logarithm as log (t); when = we set log (t) = ln(t). Our goal is to rove Theorem 1.1. Let T be a finite set of rime numbers, and let [ 2r, 2r] be a real interval such that r T 1/( 1) > 1. Then there exist infinitely many algebraic integers, all of whose conjugates in C are contained in the interval [ 2r, 2r], and all of whose conjugates in the -adic comlex numbers C (for T ) are contained in the set of -adic integers Z. (In articular, these numbers are totally real and totally -adic, for T.) Although we have stated Theorem 1.1 without reference to caacities, it is in fact caacity-theoretic. The quantity in the hyothesis is simly the caacity γ(e, { }) for E = E E, where E = [ 2r, 2r], E = Z for T, and E = Õ for T, where Õ is the ring of integers of C. The theorem is shar in the sense that if γ(e, { }) < 1 then by Fekete s theorem (see [6],. 414) the result is false. (We do not know whether or not the result holds if γ(e, { }) = 1 and T.) The roof follows the classical method of Fekete and Szegő, which involves atching a carefully chosen olynomial with real coefficients sequentially adjusting the coefficients from highest to lowest order so that they become integers in such a way that control is maintained over the
Fekete Szegő theorem 101 locations of the roots. In our case, we simultaneously atch a real olynomial, and collection of olynomials with -adic coefficients, to a common olynomial with integer coefficients. We actually rove Theorem 1.2. Under the hyotheses above, there exist monic olynomials u(x) Z[x] with distinct roots, and arbitrarily high degree, whose roots in C all belong to the real interval [ 2r, 2r], and whose roots in C all belong to Z, for each T. 2. Chebyshev olynomials and Stirling olynomials. Let r > 0 be a real number. The Chebyshev olynomials for the interval [ 2r, 2r] are defined by (2.1) T n,r (2r cos(θ)) = 2r n cos(nθ) for n = 0, 1, 2,... Clearly T n,r E = 2r n and T n,r (x) oscillates n times between ±2r n on [ 2r, 2r]. Moreover, T n,r (x) has n simle roots in [ 2r, 2r], and [ 2r, 2r] = Tn,r([ 2r 1 n, 2r n ]). It is easy to see that for n 1, T n,r (x) is a monic olynomial of degree n. Writing n T n,r (x) = z n + a k,r (n)x n k, k=1 it is shown (in Robinson [5]) that the coefficients a k,r (n) are given by 0 if k is odd, (2.2) a k,r (n) = ( 1) m r 2m n ( ) n m 1 if k = 2m is even, m m 1 where ( ) n m 1 m 1 is the binomial coefficient. In articular, for fixed k and r, a k,r (n) is itself a olynomial in n without constant term; and if r = M/N is a rational number, then the coefficients of a k,r (n) are rational numbers with denominators dividing N k k!. Thus, if r is rational, then for any fixed k 0 and any fixed integer Q 0 0, there is an integer N 0 (deending on r, k 0, and Q 0 ) such that if n is a multile of N 0 then all the a k,r (n), for 1 k k 0, are integers divisible by Q 0. The Stirling olynomial of degree n is defined by n 1 n (2.3) S n (x) = (x j) = x n + b k (n)x k. In articular S n (x) has n distinct roots in Z, for each. From the fact that ( ) x S n (x) = n! n follows S n Z = n!. k=1
102 R. Rumely Again the coefficients b k (n) (for k 1) are olynomials in n without constant term. Indeed, by a theorem of Schlömilch (see [3],. 216) (2.4) b k (n) = ( ) ( )( ) h (h j) ( 1) j+h k+h n + h 1 n + k. j h! k + h k h 0 j<h k Here each summand has an algebraic factor of n(n 1)... (n k). Since b k (n) has degree 2k and takes N to Z, the coefficients in its exansion in owers of n have denominators that divide (2k)!. It is well known that ord (n!) = (n a i )/( 1), where the a i are the base digits of n. From this we see that n (2.5) 1 log (n) ord (n!) n 1 1. 3. The basic ideas in atching. The urose of this section is to describe the atching rocess in general terms, and to rove some estimates which guide how it is carried out. Formally, the atching rocess is as follows: for each T { }, we begin with a olynomial u (0) (x) Q [z] of degree n, where n is indeendent of. The kth atching ste, for k = 1,..., n, consists of choosing numbers (k) Q and monic olynomials w (k) (x) Q [x] of degree n k, and setting (3.1) u (k) Write (x) = u (k 1) (x) + (k) w (k) (x) for each T { }. u (k) (x) = x n + n j=1 c (k),j xn j. A ste (3.1) has the effect of relacing the coefficient c (k 1),k by c (k),k = c (k 1),k + (k), leaving higher order coefficients unchanged, and modifying lower order coefficients in ways that are not imortant to us. We will choose the multiliers (k) Q so that c (k),k is a rational integer c k indeendent of. Since the kth ste only changes coefficients of terms of degree n k and lower, in the end the u (n) u(x) Z[x]. By hyothesis, (3.2) r > T (x) are all equal to the same monic olynomial 1/( 1). After shrinking r, we can assume that r = M/N is rational. Set E = [ 2r, 2r] and E = Z for T. The initial olynomials will be u (0) (x) = T n,r (x) at the archimedean lace, and u (0) (x) = S n (x) for T. These olynomials have all their roots in the sets E. The crucial
Fekete Szegő theorem 103 issue is to maintain this roerty at each ste of the atching rocess. For this to be ossible, n must be chosen large and aroriately divisible, and the atching olynomials w (k) (x) must be chosen with near-minimal su norm on E. In the case =, we will take w (k) (x) = T n k,r (x). The idea is that since u (0) (x) = T n,r (x) oscillates n times between ±2r n, if the total magnitude of the atching terms is less than 2r n then the final atched olynomial u (n) (x) will still have n roots in E. To achieve this, let h < r be such that h T 1/( 1) > 1 (cf. (3.2)). We will require that for an aroriate number L, { (k) Then for each x E, n (3.3) (k) w (k) (x) k=1 = 0 for k < L, (k) h k for k L. n h k 2r n k < k=l 1 r L (1 h/r) 2rn. If L is sufficiently large, the right side will be less than 2r n ; and then the atched olynomial will still have all its roots in [ 2r, 2r]. Requiring (k) = 0 for small k might aear to resent roblems in achieving integrality for the high-order coefficients, but our trum card is the rationality of r and the choice of n. In the case of a finite rime T, the atching olynomials w (k) (x) will be taken to be factors of the u (k 1) (x), chosen so that the cofactors f (k) (x) = u (k 1) (x)/w (k) (x) have their roots -adically distributed like the roots of a Stirling olynomial. We isolate this concet as follows. Definition 3.1. Let k be a ositive integer. A regular sequence of length k in Z is a sequence α 0, α 1,..., α k 1 Z such that for each j = 0, 1,..., k 1, ord (α j j) log (k). We simly seak of a regular sequence if k or Z is understood. Note that if α 0, α 1,..., α k 1 is a regular sequence of length k, then ord (α j j) log (k) for all j, and ord (α i α j ) < log (k) for distinct i, j. The basis for atching on the -adic side is given by Lemma 3.1. Let f(x) = k 1 (x α j) Z [x] be a olynomial whose roots form a regular sequence of length k in Z. Suose b 0, and let
104 R. Rumely Z satisfy ord ( ) k 1 + log (k) + b. Put f (x) = f(x) +. Then f (x) factors comletely over Q, and its roots αj again form a regular sequence of length k in Z. Indeed, the αj can uniquely be ut in corresondence with the α j in such a way that ord (α j α j ) log (k) + b for all j = 0,..., k 1. P r o o f. Fix one of the roots α J of f(x), and consider the Newton olygon of f (x), exanded about the oint α J. (For the theory of Newton olygons, see [1],. 37 43.) Write k f (x) = d i (x α J ) i. By assumtion d 0 =, and i=0 d 1 = ± j J(α j α J ). Since the α j form a regular sequence, if j J the ultrametric inequality imlies ord (α j α J ) = ord (j J). Therefore For i 2, so that ord (d 1 ) = ord (J!) + ord ((k 1 J)!) d i = ±d 1 = ord ((k 1)!) ord (( k 1 J j 1,...,j i 1 distinct, J )) k 1 1. [(α j1 α J )... (α ji 1 α J )] 1 ord (d i ) ord (d 1 ) (i 1) log (k). By the hyothesis on ord ( ), the Newton olygon of f (x) has a break at the oint (1, ord (d 1 )), and if its initial segment has sloe m, then m log (k)+b. Hence f (x) has a unique root αj Z for which ord (αj α J) log (k) + b. Since this holds for all J, the roots αj form a regular sequence of length k in Z. We aly Lemma 3.1 as follows. After the (k 1)st ste, write n 1 n u (k 1) (x) = (x α,j ) = z n + c,j x n j. j=1
Fekete Szegő theorem 105 Suose that α,0,..., α,k 1 form a regular sequence of length k. Then taking the atching olynomial to be and choosing (k) we obtain Z so that n 1 w (k) (x) = (x α,j ) j=k ord ( (k) ) k 1 + log (k) + b for some b 0, u (k) (x) = u (k 1) = = [ k 1 (x) + (k) w (k) (x) ] (x α,j ) + (k) w (k) k 1 n 1 (x α,j) (x α,j ). Of course, a similar result would have been obtained if any other regular sequence of roots, of length k, had been used. Note that by the corresondence between the α,j and the α,j in Lemma 3.1, there is a natural labelling of the roots of each u (k) (x) in terms of 0, 1,..., n 1. Whenever we refer to an α,j it will be using this labelling. If b is sufficiently large, then the α,j will not only form a regular sequence of length k, but they will be art of a longer regular sequence; this ermits the atching rocess to continue. However, unless b log (n), they need not be art of a regular sequence of length n; and for small k the interaction between the archimedean rime and the rimes in T will force b < log (n). In consequence, some of the roots α,j moved in early stes may stray very near to other unatched roots. These comlications account for some of the difficulties in the roof below. 4. The roof of Theorem 1.2. As noted, our goal is to merge the local olynomials u (0) (x) into a single global olynomial u(x). It is here that hyothesis (3.2) enters. Put q = T j=k 1/( 1). Since we have chosen the arameter h so that q < h < r, there is a number k 0 such that for k k 0, k/( 1)+log (k) < h k < r k. T (x)
106 R. Rumely Thus, writing c,k for the coefficient of z n k in u (k 1) (x), if k k 0 there exists a number c k Z satisfying { ck c (4.1),k < h k, ord (c k c,k ) k/( 1) + log (k) for T. Therefore, if we ut (k) = c k c,k for each, the kth ste of the atching rocess achieves an integral coefficient for x n k. For small k, (4.1) may not be satisfied: integrality for the high order coefficients must be arranged by other means. We now roceed to the details of the roof. Once and for all, fix an h with q < h < r and ut C = 1 + 2 #(T ) ln(h/q), Q =. T We can assume T is not emty; when T =, much of the argument below degenerates. In any case by [5] the result is already known in that situation. There are five stages in the atching rocess: I. Patching the coefficients c k for 1 k L 1 (n) := C ln ln(n), achieving integrality through the choice of n. II. Patching the coefficients c k for L 1 (n) < k L 2 (n) := C ln(n), at which oint h k dominates q k enough that further atching will move roots only within cosets {x Z : ord (x α,j ) > log (n)}, for each T. III. Moving the roots erturbed in stages I and II, so that for each T they are searated from other roots by at least 4C log (n) in ord value. IV. Patching the coefficients c k for 13C #(T ) L 2 (n) < k L 3 (n) := ln(n), ln(h/q) at which oint h k dominates q k so much that even the roots moved in stages I III can again be safely included in the atching rocess. V. Patching the remaining coefficients c k for L 3 (n) < k n. The atching rocess can be carried through only for certain values of n. It is sufficient to have (N1) r L 1(n) (1 h/r) > 2; (N2) ln ln(n) > (1 + 2 #(T ) ln(c) + ln(q))/ln(h/q); (N3) n > 3QL 2 (n); (N4) n e e ; (N5) n > 3QL 3 (n); (N6) L 2 (n) > 1 + max T ( 2 );
(N7) n is divisible by N L 1(n) 2L 1 (n)! T where N is the denominator of r = M/N. Fekete Szegő theorem 107 L 1(n)/( 1)+log (L 1 (n))+log (L 2 (n)), The constraints (N1) (N6) hold for all sufficiently large n. By Stirling s formula, the quantity in (N7) is easily seen to be o(n), and hence (N7) holds for infinitely many n. Stage I: Patching for 1 k < L 1 (n) = C ln ln(n). The highest-order coefficients a k (n) of u (0) (x) = T n,r (x) will be made integral through the choice of n, and, for T, the coefficients b k (n) of the u (0) (x) will be adjusted to meet them. If n satisfies (N7), then for all k L 1 (n), and all T, the a k (n) and b k (n) are rational integers satisfying (4.2) ord ( ) L 1 (n)/( 1) + log (L 1 (n)) + log (L 2 (n)). This is because the a k (n) and b k (n) are olynomials in n with rational coefficients and no constant term, and the denominators of the coefficients in both a k (n) and b k (n) divide N k (2k)!. To secify the kth atching ste, it is enough to give the target coefficient c k Z and the -adic atching olynomials w (k) (x), since (k) and w (k) (x) = T n k,r (x). We will take c k = a k (n) (so (k) n 1 w (k) (x) = (x α,j ). j=k = c k c,k = 0), and On the -adic side, using condition (4.2) and Lemma 3.1 one sees inductively that for each k L 1 (n): (a) the atching coefficients (k) satisfy (4.2), for T, and hence (b) the L 1 (n) high-order coefficients of u (k) (x) also satisfy (4.2), for T ; (c) the first L 2 (n) roots of u (k) (x) form a regular sequence of length L 2 (n) in Z. Stage II: Patching for L 1 (n) = C ln ln(n) < k L 2 (n) = C ln(n). Here both the archimedean and nonarchimedean u (k) (x) will be modified. The archimedean atching coefficients (k) = c k c,k can be chosen arbitrarily, subject to the condition (k) < h k, because under hyothesis (N1) the total archimedean atching error will be at most r n (cf. (3.3)). For L 1 (n) < k L 2 (n), we claim that (4.3) h k > T k/( 1)+log (k)+log (L 2(n)).
108 R. Rumely Indeed, (4.3) is imlied by ( ) k h > Qk #(T ) L 2 (n) #(T ) q which follows from the hyotheses on k, the definition of C, and (N2), via ( k ln(h/q) > 1 + 2 #(T ) ) ln ln(n) ln(h/q) ln(h/q) > ln(q) + 2 #(T ) ln(c) + 2 #(T ) ln ln(n) > ln(q) + #(T ) ln(k) + #(T ) ln(l 2 (n)). By (4.3), at the kth ste we can find a target coefficient c k Z such that { ck c,k < h k, ord (c k c,k ) k/( 1) + log (k) + log (L 2 (n)) for each T. As in Stage I, the -adic atching olynomials will be n 1 w (k) (x) = (x α,j ). j=k In Stage I we carefully reserved the roerty that the first L 2 (n) roots of u (k) (x) formed a regular sequence of length L 2 (n). By Lemma 3.1 and an inductive argument like that in Stage I, after each atching ste, the first L 2 (n) roots of u (k) (x) continue to form a regular sequence of length L 2 (n). Stage III. This stage only changes the -adic olynomials. For notational convenience, set m = L 2 (n) and write u (x) = u (m) (x). In Stages I and II we have adjusted the highest m coefficients of the u (x) to common integer values. However, in the rocess, we have erturbed the first m roots, and although these roots remain well-searated from each other, they may have drifted very near to other roots (they form a regular sequence of length m, but they may not be art of a regular sequence of length n). To enable the atching rocess to continue, we ause to move them away from any roots they may have strayed too near to, taking care that the m high-order coefficients of u (x) remain unchanged. This is done by means of a -adic imlicit function theorem, and involves moving a second set of m roots in a way that comensates for the first. Fix T. The coefficients of u (x) are elementary symmetric functions of the roots. Since the first m roots α,0,..., α,m 1 form a regular sequence of length m (by the construction in Stages I and II), while the last n m
Fekete Szegő theorem 109 roots are the same as those of u (0) (x) (and hence are contained in a regular sequence of length n), for each j < m there is at most one root α,µ(j) α,j such that ord (α,j α,µ(j) ) log (n) (necessarily, µ(j) m). On the other hand, since n > 3 log (m) by hyothesis (N3), it is ossible to choose a second regular sequence of length m, consisting of roots α,τ(j), 0 j < m, which avoids the delicate roots α,j and α,µ(j), 0 j < m. By our choice of the α,τ(j), for each i τ(j), 0 i < n, we have (4.4) ord (α,i α,τ(j) ) < log (n). Given vectors x, y Z m and z Z n 2m, let (x, y, z) Z n (res. (y, z) Z n m ) denote the concatenated vector, and if x = (x 1,..., x m ), let x i be x with its ith comonent omitted. Write s k (x, y, z) for the kth elementary symmetric function on the comonents of the concatenated vector. Note that for each i, (4.5) s k (x, y, z) = x i s k 1 ( x i, y, z) + s k ( x i, y, z). In consequence, as a formal derivative, x i s k (x, y, z) = s k 1 ( x i, y, z). Fixing (y, z) Z n m, consider the ma defined by S : Z m S(x) = S(x, y, z) = Z m s 1 (x, y, z) s 2 (x, y, z). s m (x, y, z). Recall that for a vector, we write ord ( ) = min(ord ( i )). Lemma 4.1. Suose m 2. Let x Z m be a vector whose comonents form a regular sequence of length m, and let Z m be a vector for which ord ( ) Then there is an x Z m 2m + b for some b 0. 1 satisfying ord (x x) m/( 1) + b such that S(x, y, z) = S(x, y, z) +.
110 R. Rumely P r o o f. We aim to aly Hensel s lemma in several variables. Consider the Jacobian matrix of S(x), which by (4.5) is 1... 1 s 1 ( x 1, y, z) s 1 ( x m, y, z) J S (x) =.. s m 1 ( x 1, y, z)... s m 1 ( x m, y, z). The usual argument for comuting Vandermonde determinants (equating variables, then examining the diagonal term), shows that det(j S (x)) = i<j(x i x j ). Similarly, if cof kl (x) denotes the (k, l)-cofactor of J S (x) (obtained from J S (x) by deleting the kth row and lth column), then cof kl (x) = ( 1) k+l (x i x j )P kl (x, y, z) i<j i,j l where P kl (x, y, z) is a olynomial with integer coefficients. Thus [J S (x) 1 ] kl = ( 1) k+1 P lk (x, y, z) 1 i m, i k (x i x k ). Since the comonents of x form a regular sequence of length m, for each k = 1,..., m, ord ( 1 i m i k ) ( (x i x k ) = ord 1 i m i k ) (i k) = ord ((k 1)!(m k)!) m 1 1. Consequently, each entry of J S (x) 1 has ord value (m 1)/( 1). The usual roof of Hensel s lemma now shows that the sequence x (i) defined by { x (0) = x, x (i+1) = x (i) J S (x (i) ) 1 [S(x (i) ) S(x) ] converges to a vector x = x ( ) with the desired roerties. Here the comonents of each x (i), and of x, form a regular sequence of length m, because the hyothesis that m 2 imlies that m 1 log (m). Note that the condition m 2 holds for all T, by (N6).
Fekete Szegő theorem 111 To move the roots α,j away from the α,µ(j), we take x = (α,τ(j) ) 0 j m 1, y = (α,j ) 0 j m 1 and let z be the vector formed from the remaining roots of u (x). Set b = log (n). Recalling that α,µ(j) is the unique root distinct from α,j such that ord (α,j α,µ(j) ) log (n) (if such a root exists), ut { α,j α,µ(j) + = 2m/( 1)+b if α,µ(j) exists, otherwise and let α,j y = (α,j) 0 j m 1. Evidently, for each j < m for which an α,µ(j) exists, we have (4.6) ord (α,j α,µ(j) ) 2m 1 + b + 1 < 4C log (n) (using the fact that m C ln(n) and ln()/( 1) < 1 for all rimes, and that C log (n) > 1 by hyothesis (N6)). Since a riori all other roots are well-searated from α,j and α,µ(j), we find that for each j < m, and all i j, (4.7) ord (α,i α,j) < 4C log (n). Set = S(x, y, z) S(x, y, z). Then ord ( ) 2m/( 1)+b, so Lemma 4.1, alied to S(x, y, z), shows there is an x with ord (x x) 2m/( 1) + b + 1 such that If we ut S(x, y, z) = S(x, y, z) + = S(x, y, z). α,τ(j) = x j, then Lemma 4.1 assures us that m ord (α,τ(j) α,τ(j)) 1 + b log (n). Thus, relacing α,τ(j) by α,τ(j) only moves the root within a coset of size ord (x) log (n), and leaves its osition in the regular sequence of length n (and hence its searation from the other roots) unchanged. In consequence, the olynomial u (x) = (x α,j) (x α,τ(j) ) (x w j ) j<m j<m
112 R. Rumely has the same m high-order coefficients as u (x), but its roots are searated from each other by at least 4C log (n) in ord value. We relace u (m) (x) by u (x). Stage IV: Patching for L 2 (n) < k L 3 (n) := 13C #(T ) ln(h/q) ln(n). The urose of this ste is to atch until the dominance of h k over q k is so great that even the delicate roots α,j and α,µ(j) with j < L 2 (n) can be safely moved. For k > L 2 (n) = C ln(n), we have (4.8) h k > T k/( 1)+log (k)+log (n). Indeed, (4.8) is imlied by ( ) k h > Qk #(T ) n #(T ) q which follows from n k > C ln(n) via ( k ln(h/q) > 1 + 2 #(T ) ) ln(n) ln(h/q) ln(h/q) > ln(n) ln(h/q) + 2 #(T ) ln(n) > ln(q) + #(T ) ln(k) + #(T ) ln(n) using ln(n) > ln ln(n) and hyothesis (N2). Put l = L 3 (n). We first choose a regular sequence of length l among the roots of u (x) which avoids the delicate roots α,j and α,µ(j) with j < L 2 (n). Since n > 3 log (l) by hyothesis (N5), and at most two of the delicate roots are searated by less than log (n) in ord value from any of the other roots, such regular sequences exist. Let one be α,λ(j), 0 j < l. Let T be the comlement of {λ(j) : 0 j < l} in {j : 0 j < n}. By construction, the delicate roots are all contained in T. For L 2 (n) < k L 3 (n), choose the target coefficients c k Z so that { ck c,k < h k, ord (c k c,k ) > k/( 1) + log (k) + log (n) for T, and atch using the olynomials l 1 w (k) (x) = (x α,λ(j) ) (x α,j ). j T j=k By Lemma 3.1, the roots α,λ(j), 0 j < k, are moved by at most log (n) in ord value, so their osition in the regular sequences (and their searation from other roots), remains unchanged.
Fekete Szegő theorem 113 Stage V: Patching for L 3 (n) < k n. In this stage, atching is carried through to the end. By Lemma 4.2 below, the delicate roots from Stages I, II and III can be safely included in the atching rocess when (4.9) h k > T k/( 1)+3(4C log (n)). A comutation like the one that gave (4.8) (using (N3), (N4), and (N5)) shows that (4.9) holds if k > L 3 (n) = 13C #(T ) ln(h/q) ln(n). Lemma 4.2. Suose f(z) = m 1 (x α,j) is a olynomial which slits over Z, and M log (m) is such that (T1) the α,j can be groued into disjoint subsets T 1,..., T l so that each T i is a subset of a regular sequence of length n i m (necessarily n i #(T i )); (T2) the α,j can be labelled so that ord (α,j j) log (n i ) if α,j T i ; (T3) ord (α,j α,k ) M for all j k. Put f (x) = f(x) +, where Z satisfies ord ( ) m + (l + 1)M. 1 Then f (x) slits comletely over Z, and its roots α,j can be uniquely set in one-to-one corresondence with the roots of f(x) in such a way that for all j, ord (α,j α,j ) > M. P r o o f. Fix a root α,j of f(x), and consider the Newton olygon of f (x) exanded about α,j : write m f (z) = d i (x α,j ) i. By assumtion, d 0 = and (4.10) d 1 = i=0 (α,j α,j ). m 1 j J We wish to estimate d 1. There is a unique integer J 0 in the range 0 J 0 < log (m) for which ord (α,j J 0 ) log (m);
114 R. Rumely in articular, ord (α,j J 0 ) log (n i ) for each i. Furthermore, by hyothesis (T1), for each T i there is at most one root α,ji T i such that ord (α,ji J 0 ) log (n i ). Let E = {j i : 1 i l} be the set of indices of these excetional roots; note that J E. If α,j is a root with j E, we claim that (4.11) ord (α,j α,j ) = min(ord (α,j j), ord (j J 0 ), ord (α,j J 0 )) = ord (j J 0 ). Indeed, if α,j T i, then ord (α,j j) log (n i ) by hyothesis (T2), while ord (α,j J 0 ) < log (n i ) since j is not excetional; and hence (4.12) ord (j J 0 ) < ord (α,j j). In articular, (4.12) imlies that j J 0. Consequently by the characterization of J 0 and the fact that j < log (m), we have ord (α,j J 0 ) log (m) but ord (α,j j) < log (m), and so (4.13) ord (j J 0 ) < ord (α,j J 0 ). Now (4.12) and (4.13) yield (4.11). On the other hand for each root α,j with j E, j J, then in any case by hyothesis (T3), (4.14) ord (α,j α,j ) M. Since J E, we have #(E\{J}) l 1. Hence by (4.10), (4.11), and (4.14), ( m 1 ) ( ord (d 1 ) = ord (j J 0 ) + ord j E ( m 1 ord (j J 0 ) j J 0 There are now two cases. If J 0 < m, then ( m 1 ord (j J 0 ) j J 0 ) j E j J ) + (l 1)M. ) (α,j α,j ) = ord (J 0!) + ord ((m J 0 1)!) m 1 1. However, if J 0 m, then by the bound J 0 m < J 0 < log (m) we have
Fekete Szegő theorem 115 log (J 0 m) log (m), and so by (2.5) it follows that ( m 1 ord (j J 0 ) j J 0 ) = ord (J 0!) ord ((J 0 m)!) J ( ) 0 1 1 J0 m 1 log (J 0 m) = m 1 1 + log (m). By assumtion M log (m). Thus, in either case, (4.15) ord (d 1 ) m 1 1 + lm. For the coefficients d i with i 2, we have d i = ±d 1 [(α,k1 α,j )... (α,ki 1 α,j )] 1 so that k 1,...,k i 1 distinct, J ord (d i ) ord (d 1 ) (i 1)M. By our hyothesis on ord ( ), the Newton olygon of f (x) has a break at the oint (1, ord (d 1 )), and if its initial segment has sloe m, then m > M. Hence, f (x) has a unique root α J for which ord (α J α J) > M. By the uniqueness, this root belongs to Q, and hence to Z. At the kth ste of the atching rocess, we aly Lemma 4.2 with m = k, l = 2, M = 4C log (n) and k 1 f(x) = (x α,j ) where the roots α,j are those of the current olynomial u (k 1) (x), given their natural labelling. Clearly M log (m). We will take T 1 = {α,j : 0 j < L 2 (n) }, T 2 = {α,j : L 2 (n) j < k}, with m 1 = L 2 (n) and m 2 = k; thus T 1 consists of the roots moved in Stages I and II, and T 2 is an initial segment of the remaining roots. In Stages III, IV, and V, roots are moved only by quantities with ord value log (n), reserving their osition in the regular sequence of length n. Hence T 2 is a subset of a regular sequence of length n, but we only use that it is a subset of a regular sequence of length k. Hyothesis (T2) in Lemma 4.2 is satisfied for T 1, since T 1 is regular of length m 1 by the construction in Stages
116 R. Rumely I and II; and it is satisfied for T 2 since the original labelling of the roots was such that ord (α,j j) log (n) for roots in T 2, and this roerty was reserved thoughout the atching rocess. As noted above, for k > L 3 (n) = 13C #(T ) ln(h/q) ln(n) we have h k > T k/( 1)+3(4C log (n)). We can thus find target coefficients c k Z such that { ck c,k < h k, ord (c k c,k ) > k/( 1) + 3 4C log (n) for T. The atching olynomials will simly be n 1 w (k) (x) = (x α,j ) for k < n, j=k w (n) (x) = 1. It follows from Lemma 4.2 that at each ste the roots are moved by quantities with ord value > 4C log (n). Hence they remain searated in ord value by at least 4C log (n), and the hyotheses of Lemma 4.2 continue to hold; the atching rocess carries through to the end. References [1] E. A r t i n, Algebraic Numbers and Algebraic Functions, Gordon and Breach, Science Publishers, New York, 1967. [2] D. Cantor, On an extension of the definition of transfinite diameter and some alications, J. Reine Angew. Math. 316 (1980), 160 207. [3] L. Comtet, Advanced Combinatorics, Reidel, Boston, 1979. [4] M. Fekete and G. Szegő, On algebraic equations with integral coefficients whose roots belong to a given set, Math. Z. 63 (1955), 158 172. [5] R. M. Robinson, Conjugate algebraic integers in real oint sets, Math. Z. 84 (1964), 415 427. [6] R. Rumely, Caacity Theory on Algebraic Curves, Lecture Notes in Math. 1378, Sringer, New York, 1989. Deartment of Mathematics University of Georgia Athens, GA 30602, U.S.A. E-mail: rr@alha.math.uga.edu Received on 27.4.1998 (3369)