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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...
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]
Polynomials with integer coefficients and their zeros
"... Abstract. We study several related problems on polynomials with integer coefficients. This includes the integer Chebyshev problem, and the Schur problems on means of algebraic numbers. We also discuss interesting applications to the approximation by polynomials with integer coefficients, and to the ..."
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Abstract. We study several related problems on polynomials with integer coefficients. This includes the integer Chebyshev problem, and the Schur problems on means of algebraic numbers. We also discuss interesting applications to the approximation by polynomials with integer coefficients, and to the growth of coefficients for polynomials with roots located in prescribed sets. The distribution of zeros for polynomials with integer coefficients plays an important role in all of these problems.
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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A number of open problems on constructive function theory are presented. These were submitted by participants of Constructive Function Theory Tech04.