M. Fekete, Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koe#zienten,Math.Zeit.17 (1923), 228--249.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....we observe that on intervals [a, b] of length greater than or equal to 4 we have n [a, b] # [a, b] 1. We will thus from now on restrict our attention to intervals of length at most 4. Hilbert [12] showed that there exists an absolute constant c so that L2 [a,b] cn and Fekete [9] showed that 1 2 n 1 n 2 . For refinements of their inequalities, see Kashin [13] From the above it follows that (1.8) 4 # Recall that b 4. There is a pretty argument due to Gelfond [see 10] to see that integer coe#cients really are a restriction on [0, 1] If 0 ....

M. Fekete, Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koe#zienten,Math.Zeit.17 (1923), 228--249.

....Vandermonde determinant (1.10) It is known that x1 , x n#[a,b] V n (x 1 , x n ) cap( a, b] w) see Theorem III.1.3 of [25] The quantity on the left hand side of (2. 17) is called the weighted transfinite diameter of [a, b] In the case w 1, it was introduced by Fekete [12] for arbitrary compact sets in the plane. Szego [28] showed that the transfinite diameter coincides with the logarithmic capacity, so that (2.17) is a generalization of his result. We quantify the rate of convergence in (2.17) Lemma 2.5. Let w be as in (2.1) There exist constants d = d(w) 1 ....

M. Fekete, Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koe#zienten, Math. Zeit. 17 (1923), 228-249.

....observation leads us to the following consequences. Claim 5.1 For k 3 and all m; n and s one has f k;s (n m) f k;s (n) f k;s (m 1) 1 : Claim 5.2 For every pair k; s the limit ( nite or in nite) lim n 1 f k;s (n) n exists. Proof. Follows by a standard application of Fekete s lemma [9]. The proof for the case s = k 1 is presented in the monograph [11] of Jensen and Toft, pp. 100 101) Indeed, for a xed pair k; s denote h(n) f k;s (n 1) then from Claim 5.1 h(n m) f k;s (n m 1) f k;s (n 1) f k;s (m 1) h(n) h(m) Thus h(n) is submodular and by Fekete s ....

M. Fekete,  Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koecienten, Math. Z. 17 (1923), 228-249.

....SMALL POLYNOMIALS WITH INTEGER COEFFICIENTS 3 sets of conjugate algebraic integers. Thus one may be able to explicitly find those polynomials with integer coe#cients and all roots in [a, b] b a 4. These results were generalized to the case of an arbitrary compact set E # C by Fekete [9], who developed a new analytic setting for the problem, by introducing the transfinite diameter of E and showing that it is equal to t C (E) Both quantities were later proved to be equal to the logarithmic capacity cap(E) by Szego [41] Therefore we state the result of Fekete as follows: 1.10) ....

M. Fekete, Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koe#zienten, Math. Zeit. 17 (1923), 228-249.

....length less than 4, can contain only nitely many complete sets of conjugate algebraic integers. Thus one may be able to explicitly nd those polynomials with integer coecients and all roots in [a; b] b a 4. These results were generalized to the case of an arbitrary compact set E C by Fekete [9], who developed a new analytic setting for the problem, by introducing the trans nite diameter of E and showing that it is equal to t C (E) Both quantities were later proved to be equal to the logarithmic capacity cap(E) by Szeg o [41] Therefore we state the result of Fekete as follows: 1.10) ....

M. Fekete,  Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koezienten, Math. Zeit. 17 (1923), 228-249.

....the above limit exists (cf. 14, Ch. 10] or [3] Observe from (1.1) and (1.3) that if b Gamma a 4 then q n (x) j 1 for any n 2 N and inch( a; b] 1. However, if b Gamma a 4 then b Gamma a 4 = cheb( a; b] inch( a; b] 1. 5) On the other hand, the results of Hilbert [10] and Fekete [5] imply that inch( a; b] s b Gamma a 4 (1.6) see [3] The exact value of the integer Chebyshev constant and an explicit (or even asymptotic) form of the integer Chebyshev polynomials is not known for any [a; b] with b Gammaa 4. Perhaps the most studied case, due to the interest in the ....

M. Fekete, Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Zeit. 17 (1923), 228-249. CHEBYSHEV POLYNOMIALS WITH INTEGER COEFFICIENTS 13

....F becomes unique and can be computed effectively using, for example, Grobner basis algorithms (see e.g. BM82] BW91] MMM] For n = 1 there are formulas for interpolating polynomials due to Newton, Lagrange and Hermite. These formulas have been applied in the theory of approximation (see e.g. [Fe23], Le54] Wi70] The examination of the interpolation formula s structure has led to rather reach theories of polynomial interpolation for several variables and their application to the theory of approximation (see e.g. Si62] Wi73] Br97] In particular Newton s divided differences ....

M. Fekete, Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koefficienten. Math. Zeitschr. 17, pp. 228--249, 1923.

....(x a ) x 1 ) m and therefore x a m; similarly x k m so that (x k ) m, hence (x b ) x k ) m and therefore x b m, contradicting b a. Recalling that k precedes 1 in oe, the identity permutation in S n avoids oe and demonstrates that F (n; oe) 1 for every n 1. Fekete s lemma [4], see also [9] is that if a 1 ; a 2 ; 2 R satisfy for all m; n 1, am a n am n , then lim n 1 a n =n = inf n1 a n =n 2 [ Gamma1; 1) Applying this with a n : Gamma log F (n; oe) completes our proof. There exist [10] examples with oe; oe 0 2 S k , with oe 0 the identity permutation, ....

Fekete, M. (1923) Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Z. 17, 228-249.

....(x 1 ) # m and therefore x a # m; similarly x k mso that # (x k ) m, hence # (x b ) # # (x k ) m and therefore x b m, contradicting b a. Recalling that k precedes 1 in #, the identity permutation in S n avoids # and demonstrates that F (n, #) # 1 for every n # 1. Fekete s lemma [4], see also [9] is that if a 1 ,a 2 , # R satisfy for all m,n # 1, am a n # am n , then lim n## a n n = inf n#1 a n n # [ #,#) Applying this with a n : log F (n, #) completes our proof. There exist [10] examples with #, # # #S k ,with# # the identity permutation, and F (n, ....

Fekete, M. (1923) Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koe#zienten. Math. Z. 17, 228-249.

....a ) x 1 ) m and therefore x a m; similarly x k m so that (x k ) m, hence (x b ) x k ) m and therefore x b m, contradicting b a. Recalling that k precedes 1 in , the identity permutation in S n avoids and demonstrates that F (n; 1 for every n 1. Fekete s lemma [4], see also [9] is that if a 1 ; a 2 ; 2 R satisfy for all m; n 1, am a n am n , then lim n 1 a n =n = inf n 1 a n =n 2 [1;1) Applying this with a n : log F (n; completes our proof. There exist [10] examples with ; 0 2 S k , with 0 the identity permutation, and F (n; ....

Fekete, M. (1923)  Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen KoeÆzienten. Math. Z. 17, 228-249.

.... and that we must content ourselves with the weaker statement v(P Omega P 0 ) v(P )v(P 0 ) If we define P Omega n as P Omega P Omega : Omega P with n occurrences of P , then this inequality implies that v(P Omega i j ) v(P Omega i )v(P Omega j ) for all i; j; by Fekete s lemma [2], we conclude that as n gets large the quantity n q v(P Omega n ) approaches its infimum, which we call the asymptotic optimum value of P . The following theorem gives conditions on P that force the asymptotic optimum value to equal the value of the LP relaxation of P . Theorem 1: Let P = ....

F. M. Fekete, Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Zeit. 17 (1923), 228-249.

....restricted to a few domains like the interval and its tensor products. In this work we propose a new spectral element method for both triangles and quadrilaterals which retains the diagonal mass matrix. Instead of being based on Gauss Lobatto quadrature points, it will be based on Fekete points [13]. Other references for Fekete points can be found in the recent paper [5] which summarizes many of the known results and gives some open questions. The references [6,23,1] represent the only work we are aware of in which Fekete points are computed in more than one dimension. A similar approach, ....

M. Fekete, Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Zeits. 17, (1923).

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M. Fekete, Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z. 17 (1923), 228--249.

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