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The Integer Chebyshev Problem
, 1995
"... . We are concerned with the problem of minimizing the supremum norm on an interval of a nonzero polynomial of degree at most n with integer coefficients. This is an old and hard problem that cannot be exactly solved in any nontrivial cases. We examine the case of the interval [0; 1] in most detail ..."
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. We are concerned with the problem of minimizing the supremum norm on an interval of a nonzero polynomial of degree at most n with integer coefficients. This is an old and hard problem that cannot be exactly solved in any nontrivial cases. We examine the case of the interval [0; 1] in most detail. Here we improve the known bounds a small but interesting amount. This allows us to garner further information about the structure of such minimal polynomials and their factors. This is primarily a (substantial) computational exercise. We also examine some of the structure of such minimal "integer Chebyshev" polynomials, showing for example, that on small intevals [0; ffi] and for small degrees d, x d achieves the minimal norm. There is a natural conjecture, due to the Chudnovskys' and others, as to what the "integer transfinite diameter" of [0; 1] should be. We show that this conjecture is false. The problem is then related to a trace problem for totally positive algebraic integers due t...
Small Polynomials With Integer Coefficients
"... this paper is the study of polynomials with integer coe#cients that minimize the sup norm on the set E. In particular, we consider the asymptotic behavior of these polynomials and of their zeros. Let P n (C) and P n (Z) be the classes of algebraic polynomials of degree at most n, respectively wi ..."
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this paper is the study of polynomials with integer coe#cients that minimize the sup norm on the set E. In particular, we consider the asymptotic behavior of these polynomials and of their zeros. Let P n (C) and P n (Z) be the classes of algebraic polynomials of degree at most n, respectively with complex and with integer coe#cients. The problem of minimizing the uniform norm on E by monic polynomials from P n (C) is well known as the Chebyshev problem (see [4], [31], [43], [16], etc.) In the classical case E = [1, 1], the explicit solution of this problem is given by the monic Chebyshev polynomial of degree n: T n (x) := 2 1n cos(n arccos x), n # N. Using a change of variable, we can immediately extend this to an arbitrary interval [a, b] # R, so that t n (x) := # b  a 2 # n T n # 2x  a  b b  a # is a monic polynomial with real coe#cients and the smallest uniform norm on [a, b] among all monic polynomials from P n (C). In fact, (1.1) #t n # [a,b] = 2 # b  a 4 # n , n # N, and we find that the Chebyshev constant for [a, b] is given by (1.2) t C ([a, b]) := lim n## #t n # 1/n [a,b] = b  a 4 . The Chebyshev constant of an arbitrary compact set E # C is defined in a similar fashion: (1.3) t C (E) := lim n## #t n # 1/n E , where t n is the Chebyshev polynomial of degree n on E. It is known that t C (E) is equal to the transfinite diameter and the logarithmic capacity cap(E) of the set E (cf. [43, pp. 7175], [16] and [30] for the definitions and background material). 2000 Mathematics Subject Classification. Primary 11C08, 30C10; Secondary 31A05, 31A15. Key words and phrases. Chebyshev polynomials, integer Chebyshev constant, integer transfinite diameter, zeros, multiple factors, asymptotic...
The Gelfond–Schnirelman method in prime number theory
 Canad. J. Math
, 2005
"... Abstract. The original Gelfond–Schnirelman method, proposed in 1936, uses polynomials with integer coefficients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lowe ..."
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Abstract. The original Gelfond–Schnirelman method, proposed in 1936, uses polynomials with integer coefficients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for the integral of Chebyshev’s ψfunction, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomialtype weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support. 1 Lower Bounds for Arithmetic Functions Let π(x) be the number of primes not exceeding x. The celebrated Prime Number Theorem (PNT), suggested by Legendre and Gauss, states that (1.1) π(x) ∼ x log x as x → ∞. We include a very brief sketch of its history, referring for details to many excellent books and surveys available on this subject (see, e.g., [8, 10, 17, 29]). Chebyshev [6] made the first important step towards the PNT in 1852, by proving the bounds (1.2) 0.921 x log x
The monic integer transfinite diameter
 Math. Comp
, 2006
"... Abstract. We study the problem of finding nonconstant monic integer polynomials, normalized by their degree, with small supremum on an interval I. The monic integer transfinite diameter tM(I) is defined as the infimum of all such supremums. We show that if I has length 1 then tM(I) = 1 2. We make t ..."
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Abstract. We study the problem of finding nonconstant monic integer polynomials, normalized by their degree, with small supremum on an interval I. The monic integer transfinite diameter tM(I) is defined as the infimum of all such supremums. We show that if I has length 1 then tM(I) = 1 2. We make three general conjectures relating to the value of tM(I) for intervals I of length less that 4. We also conjecture a value for tM([0, b]) where 0 < b ≤ 1. We give some partial results, as well as computational evidence, to support these conjectures. We define functions L−(t) and L+(t), which measure properties of the lengths of intervals I with tM(I) on either side of t. Upper and lower bounds are given for these functions. We also consider the problem of determining tM(I) when I is a Farey interval. We prove that a conjecture of Borwein, Pinner and Pritsker concerning this value is true for an infinite family of Farey intervals.
Open problems on constructive function theory
, 2006
"... A number of open problems on constructive function theory are presented. These were submitted by participants of Constructive Function Theory Tech04. ..."
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Cited by 3 (1 self)
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A number of open problems on constructive function theory are presented. These were submitted by participants of Constructive Function Theory Tech04.
DISTRIBUTION OF PRIMES AND A WEIGHTED ENERGY PROBLEM
"... Abstract. We discuss a recent development connecting the asymptotic distribution of prime numbers with weighted potential theory. These ideas originated with the GelfondSchnirelman method (circa 1936), which used polynomials with integer coefficients and small sup norms on ¢ £¥¤§¦© ¨ to give a Cheb ..."
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Abstract. We discuss a recent development connecting the asymptotic distribution of prime numbers with weighted potential theory. These ideas originated with the GelfondSchnirelman method (circa 1936), which used polynomials with integer coefficients and small sup norms on ¢ £¥¤§¦© ¨ to give a Chebyshevtype lower bound in prime number theory. A generalization of this method for polynomials in many variables was later studied by Nair and Chudnovsky, who produced tight bounds for the distribution of primes. Our main result is a lower bound for the integral of Chebyshev’s �function, expressed in terms of the weighted capacity for polynomialtype weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support. This new connection leads to some interesting open problems on weighted capacity.
THE MULTIVARIATE INTEGER CHEBYSHEV PROBLEM
"... Abstract. The multivariate integer Chebyshev problem is to find polynomials with integer coefficients that minimize the supremum norm over a compact set in C d. We study this problem on general sets, but devote special attention to product sets such as cube and polydisk. We also establish a multivar ..."
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Abstract. The multivariate integer Chebyshev problem is to find polynomials with integer coefficients that minimize the supremum norm over a compact set in C d. We study this problem on general sets, but devote special attention to product sets such as cube and polydisk. We also establish a multivariate analog of the HilbertFekete upper bound for the integer Chebyshev constant, which depends on the dimension of space. In the case of single variable polynomials in the complex plane, our estimate coincides with the HilbertFekete result. 1. The integer Chebyshev problem and its multivariate counterpart The supremum norm on a compact set E ⊂ C d, d ∈ N, is defined by ‖f‖E: = sup f(z). z∈E We study polynomials with integer coefficients that minimize the sup norm on a set E, and investigate their asymptotic behavior. The univariate case (d = 1) has a long history, but the problem is virtually untouched for d ≥ 2. Let Pn(C) and Pn(Z) be the classes of algebraic polynomials in one variable, of degree at most n, respectively with complex and with integer coefficients. The problem of minimizing the uniform norm on E by monic polynomials from Pn(C) is the classical Chebyshev problem (see [5], [22], [26], etc.) For E = [−1, 1], the explicit solution of this problem is given by the monic Chebyshev polynomial of degree n: Tn(x): = 2 1−n cos(n arccos x), n ∈ N. By a linear change of variable, we immediately obtain that ( ) n () b − a 2x − a − b tn(x): = Tn 2 b − a is a monic polynomial with real coefficients and the smallest norm on [a, b] ⊂ R among all monic polynomials of degree n from Pn(C). In fact, ( ) n b − a (1.1) ‖tn ‖ [a,b] = 2, n ∈ N, 4 and the Chebyshev constant for [a, b] is given by 1/n b − a (1.2) tC([a, b]): = lim ‖tn‖ n→ ∞ [a,b]
GENERALIZED GORSHKOVWIRSING POLYNOMIALS AND THE INTEGER CHEBYSHEV PROBLEM
"... Abstract. The Integer Chebyshev Problem is the problem of finding an integer polynomial of degree n such that the supremum norm on [0, 1] is minimized. The most common technique used to find upper bounds is by explicit construction of an example. This is often (although not always) done by heavy c ..."
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Abstract. The Integer Chebyshev Problem is the problem of finding an integer polynomial of degree n such that the supremum norm on [0, 1] is minimized. The most common technique used to find upper bounds is by explicit construction of an example. This is often (although not always) done by heavy computational use of the LLL and the Simplex method. Among the first methods developed to find lower bounds was through a sequence of polynomials known as the GorshkovWirsing polynomials. This paper studies properties of the GorshkovWirsing polynomials. It is shown how to construct generalized GorshkovWirsing polynomials on any interval [a, b], with a, b ∈ Q. An extensive search for generalized GorshkovWirsing polynomials is done for a large family of [a, b]. Using generalized GorshkovWirsing polynomials, LLL and the Simplex method, upper and lower bounds for the Integer Chebyshev Constant on intervals other than [0,1] are calculated. These methods are compared with other existing methods. 1.
Article electronically published on January 8, 2003 MONIC INTEGER CHEBYSHEV PROBLEM
"... 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 ∈Mn(Z) satisfies ‖Mn‖E = inf Pn∈Mn(Z) ‖Pn‖E. and the monic integer Cheb ..."
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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 ∈Mn(Z) satisfies ‖Mn‖E = inf Pn∈Mn(Z) ‖Pn‖E. and the monic integer Chebyshev constant is then defined by tM (E): = lim n→ ∞ ‖Mn‖1/n 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 consecutive 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