Emiliano Aparicio Bernardo, On the asymptotic structure of the polynomials of minimal diophantic deviation from zero, J. Approx. Theory 55 (1988), 270-278. Home/Search Document Not in Database Summary Related Articles Check |
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Small Polynomials With Integer Coefficients - Pritsker (Correct)
....diameter, zeros, multiple factors, asymptotics, potentials, weighted polynomials. Research supported in part by the National Science Foundation grant DMS 9996410. c #1997 American Mathematical Society 1 2 IGOR E. PRITSKER One may notice that the Chebyshev polynomials on the interval [ 2, 2] have integer coe#cients. The roots of the n th Chebyshev polynomial on [ 2, 2] are (1.4) x k = 2 cos (2k 1)# 2n , k = 1, n. A remarkable result of Kronecker [21] states that any complete set of conjugate algebraic integers, i.e. roots of a monic irreducible polynomial over Z, all ....
....polynomials. Research supported in part by the National Science Foundation grant DMS 9996410. c #1997 American Mathematical Society 1 2 IGOR E. PRITSKER One may notice that the Chebyshev polynomials on the interval [ 2, 2] have integer coe#cients. The roots of the n th Chebyshev polynomial on [ 2, 2] are (1.4) x k = 2 cos (2k 1)# 2n , k = 1, n. A remarkable result of Kronecker [21] states that any complete set of conjugate algebraic integers, i.e. roots of a monic irreducible polynomial over Z, all contained in [ 2, 2] must belong to one of the sets (1.4) for some n # N. ....
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Emiliano Aparicio Bernardo, On the asymptotic structure of the polynomials of minimal diophantic deviation from zero, J. Approx. Theory 55 (1988), 270-278.
Small Polynomials With Integer Coefficients - Pritsker (2000) (Correct)
....and compute their norms, to nd out that this is quite a nontrivial exercise. It was noticed in many papers that small polynomials from Pn (Z) n 2 N; arise as products of powers of polynomials from Fn ; k n: Aparicio was the rst to prove this in the following strong form (cf. Theorem 3 in [2]) If a sequence Qn 2 Pn (Z) n 2 N; satis es (1.19) lim n 1 kQn k 1=n [0;1] t Z ( 0; 1] then (1.20) Qn (x) x(1 x) 1 n] 2x 1) 2n] 5x 2 5x 1) 3n] Rn (x) as n 1; where (1.21) 1 0:1456; 2 0:0166 and 3 0:0037; and Rn 2 Pn (Z) n 2 N. This gives a good ....
....( 2 1 2 1) F w U w (0) 0:179335 and 4 2 1 2 1 exp ( 2 1 2 1) F w U w (1=4) 0:179335: 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.08 0.1 0.12 0.14 0.16 0.18 G Figure 1. Region G for 1 and 2 . Note that, in addition to improving the previous lower bounds obtained in [2], 5] and [14] 3.11) also gives the upper bounds for 1 and 2 . 3.2. Three and more factors on [0; 1=4] Numerical approach. It is natural to expect improvements in the bounds for t Z ( 0; 1=4] and for i s, if we use three or more known factors from (3.4) There is, however, a substantial ....
Emiliano Aparicio Bernardo, On the asymptotic structure of the polynomials of minimal diophantic deviation from zero, J. Approx. Theory 55 (1988), 270-278.
Chebyshev Polynomials With Integer Coefficients - Pritsker (Correct)
....4 IGOR E. PRITSKER 2. Asymptotic structure of the integer Chebyshev polynomials We are interested in the asymptotic structure of the polynomials Q n 2 P n (Z) satisfying kQ n k [0;1] inf 06jpn2Pn (Z) kp n k [0;1] n 2 N: 2.1) This problem was originally proposed by A. O. Gelfond (cf. [1]) It is known that the polynomials Q n satisfying (2.1) have factors that tend to repeat and to increase in power as n 1 (see [14, Ch. 10] and [3] for a discussion) In particular, Aparicio (cf. Theorem 3 in [1] showed that if fQ n g 1 n=1 ae P n (Z) satisfy (2.1) then Q n (x) x(1 Gamma ....
....n k [0;1] n 2 N: 2.1) This problem was originally proposed by A. O. Gelfond (cf. 1] It is known that the polynomials Q n satisfying (2.1) have factors that tend to repeat and to increase in power as n 1 (see [14, Ch. 10] and [3] for a discussion) In particular, Aparicio (cf. Theorem 3 in [1]) showed that if fQ n g 1 n=1 ae P n (Z) satisfy (2.1) then Q n (x) x(1 Gamma x) ff 1 n] 2x Gamma 1) ff 2 n] 5x 2 Gamma 5x 1) ff 3 n] R n (x) as n 1; 2.2) where ff 1 0:1456; ff 2 0:0166 and ff 3 0:0037; 2.3) and R n 2 P n (Z) n 2 N. Borwein and Erd elyi proved ....
Emiliano Aparicio Bernardo, On the asymptotic structure of the polynomials of minimal diophantic deviation from zero, J. Approx. Theory 55 (1988), 270-278.
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