MATHEMATICS OF COMPUTATION Volume 72, Number 244, Pages 1901{1916 S 0025-5718(03)01477-7 Article electronically published on January 8, 2003 MONIC INTEGER CHEBYSHEV PROBLEM P. B. BORWEIN, C. G. PINNER, AND I. E. PRITSKER Abstract. We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let Mn(Z) denote the monic polynomials of degree n with integer coefficients. A monic integer Chebyshev polynomial Mn 2Mn(Z)satisfies kMnkE =infkPnkE : Pn2Mn(Z) and the monic integer Chebyshev constant is then defined by 1=n tM (E) := lim kMnk : n!1 E This is the obvious analogue of the more usual integer Chebyshev constant that has been much studied. We compute tM (E) for various sets, including all finite sets of rationals, and make the following conjecture, which we prove in many cases. Conjecture. Suppose [a2=b2;a1=b1] is an interval whose endpoints are con- secutive Farey fractions. This is characterized by a1b2 − a2b1 =1: Then tM [a2=b2;a1=b1]=max(1=b1; 1=b2): This should be contrasted with the nonmonic integer Chebyshev constant case, where the only intervals for which the constant is exactly computed are intervals of length 4 or greater. 1. Introduction and general results Define the uniform (sup) norm on a compact set E ⊂ C by k k j j f E := sup f(z) : z2E We study the monic polynomials with integer coefficients that minimize the sup norm on the set E.LetPn(C)andPn(Z) be the classes of algebraic polynomials of degree at most n, respectively with complex and with integer coefficients. Similarly, we define the classes of monic polynomials Mn(C)andMn(Z)ofexact degree n 2 N. The problem of minimizing the uniform norm on E by polynomials from Mn(C) is well known as the Chebyshev problem (see [1], [15], [17], [8], etc.). In the classical case, E =[−1; 1], the explicit solution of this problem is given by the Received by the editor August 29, 2001 and, in revised form, December 20, 2001. 2000 Mathematics Subject Classification. Primary 11C08; Secondary 30C10. Key words and phrases. Chebyshev polynomials, integer Chebyshev constant, integer transfi- nite diameter. Research of the authors was supported in part by the following grants: NSERC of Canada and MITACS (Borwein), NSF grant EPS-9874732 and matching support from the state of Kansas (Pinner), and NSF grant DMS-9996410 (Pritsker). c 2003 by the authors 1901 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1902 P. B. BORWEIN, C. G. PINNER, AND I. E. PRITSKER monic Chebyshev polynomial of degree n: 1−n Tn(x):=2 cos(n arccos x);n2 N: Using a change of variable, we can immediately extend this to an arbitrary interval [a; b] ⊂ R,sothat b − a n 2x − a − b t (x):= T n 2 n b − a is a monic polynomial with real coefficients and the smallest uniform norm on [a; b] among all polynomials from M (C). In fact, n b − a n (1.1) kt k =2 ;n2 N; n [a;b] 4 and we find that the Chebyshev constant for [a; b]isgivenby 1=n b − a (1.2) tC([a; b]) := lim ktnk = : n!1 [a;b] 4 The Chebyshev constant of an arbitrary compact set E ⊂ C is defined in a similar fashion: 1=n (1.3) tC(E) := lim ktnk ; n!1 E where tn is the Chebyshev polynomial of degree n on E (the monic polynomial of exact degree n of minimal supremum norm on E). It is known that tC(E)isequal to the transfinite diameter and the logarithmic capacity cap(E)ofthesetE (cf. [17, pp. 71-75], [8] and [14] for the definitions and background material). An integer Chebyshev polynomial Qn 2Pn(Z)foracompactsetE ⊂ C is defined by (1.4) kQnkE =infkPnkE; 06≡Pn2Pn(Z) where the inf is taken over all polynomials from Pn(Z) which are not identically zero. Further, the integer Chebyshev constant (or integer transfinite diameter) for E is given by 1=n (1.5) tZ(E) := lim kQnk : n!1 E The integer Chebyshev problem is also a classical subject of analysis and number theory (see [11, Ch. 10], [3], [2], [6], [7], [9], [16], [13] and the references therein). It does not require the polynomials to be monic. We define the associated quantities for the monic integer Chebyshev problem as follows. A monic integer Chebyshev polynomial Mn 2Mn(Z); deg Mn = n; satisfies (1.6) kMnkE =infkPnkE: Pn2Mn(Z) The monic integer Chebyshev constant is then defined by 1=n 1=n (1.7) tM (E) := lim kMnk =infkMnk ; n!1 E n E where the existence of this limit and the last equality follows by a standard argument presented in Lemma 3.1. The monic integer Chebyshev problem is quite different from the classical integer Chebyshev problem, as we show in this paper. It is immediately clear from the definitions (1.4)-(1.7) that (1.8) tM (E) ≥ tZ(E): License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use MONIC INTEGER CHEBYSHEV PROBLEM 1903 Note that, for any Pn 2Pn(Z), kPnkE = kPnkE∗ ; ∗ where E := E [fz :¯z 2 Eg, because Pn has real coefficients. Thus the (monic) integer Chebyshev problem on a compact set E is equivalent to that on E∗,and we can assume that E is symmetric with respect to the real axis (R-symmetric) without loss of generality. Our first result shows that the monic integer Chebyshev constant coincides with the regular Chebyshev constant (capacity) for sufficiently large sets. Theorem 1.1. If E is R-symmetric and cap(E) ≥ 1,then (1.9) tM (E)=cap(E): ≥ k k ≥ We remark that tZ(E)=1forthesetsE with cap(E) 1: Indeed, Pn E n (cap(E)) for any Pn 2Pn(Z) of exact degree n (cf. [14, p. 155]). Thus Qn(z) ≡ 1 is a minimizer for (1.4) in this case. An argument going back to Kakeya (cf. [12] or [16]) gives Proposition 1.2. Let E ⊂ C be a compact R-symmetric set. If cap(E) < 1,then tM (E) < 1: We show below that this statement cannot be significantly improved. The monic integer Chebyshev constant shares a number of standard properties with tZ(E)andtC(E), such as the monotonicity property below. Proposition 1.3. Let E ⊂ F ⊂ C.Then tM (E) ≤ tM (F ): Another generic property of importance is the following (see [5] and Theorem 2 of [8, Sect. VII.1]). ⊂ C −1 Proposition 1.4. Let E be a compact set. If Pn (E) is the inverse image of E under Pn 2Mn(Z); deg Pn = n; then −1 1=n (1.10) tM Pn (E) =(tM (E)) : Perhaps, the most distinctive feature of tM (E) is that it may be different from zero even for a single point. For example (see section 2 below), suppose that m; n 2 Z; where n ≥ 2and(m; n)=1: Then n o m 1 (1.11) t = : M n n On the other hand, if a 2 R is irrational, then (1.12) tM (fag)=0: n This result has several interesting consequences. Consider En := fz : z = 1=2g;n2 N: It is obvious that cap(En)=tC(En) = 0 for any n 2 N: How- −1=n ever, (1.11) and (1.10) imply that tM (En)=2 ! 1; as n !1: Thus no uniform upper estimatep of tM (E) in terms of cap(E) is possible, in contrast with the inequality tZ(E) ≤ cap(E)p (see the results ofp Hilbert [10] and Fekete [4]). f g {− g Wep also note thatp tM ( 1p= 2 )=tM ( 1= 2 ) = 0 by (1.12), while tM (f1= 2g[f−1= 2g)=1= 2 by (1.10) and (1.11). This shows that another License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1904 P. B. BORWEIN, C. G. PINNER, AND I. E. PRITSKER well known property of capacity is not valid for tM (E): Namely, capacity (Cheby- shev constant) for the union of two sets of zero capacity is still zero (cf. Theorem III.8 of [17, p. 57]). Combining Proposition 1.3, Proposition 1.4 and (1.11), one can find the explicit values of the monic integer Chebyshev constant for many intervals and other sets. Theorem 1.5. Let n 2 Z: The following relations hold true: 1 n − 1 1 1 1 (1.13) t 0; = t ; 1 = t − ; = ;n≥ 2; M n M n M n n n s 1 1 1 1 (1.14) t −p ; p = t 0; = p ;n≥ 2; M n n M n n 1 1 1 1 (1.15) t n; n + = t n − ;n = t 0; = ; M 2 M 2 M 2 2 s 1 1 (1.16) t ([n; n +1])=t ([0; 1]) = t 0; = M M M 4 2 and p 1 (1.17) tM ([n; n +2])=tM ([−1; 1]) = tM ([0; 1]) = p : 2 p p Also, if E ⊂ [(1 − 2)=2; (1 + 2)=2] and f1=2g2E,then " p p #! 1 − 2 1+ 2 1 1 (1.18) t (E)=t ; = t = : M M 2 2 M 2 2 Of course, the above list of values can be extended further. It is worth mentioning that finding the value of tZ([0; 1]) is a notoriously difficult problem, where we do not even have a current conjecture (see [3], [11, Ch.