Geodesic
Chebyshev Polynomials and Integer Coefficients
Matematičeskie zametki, Tome 105 (2019) no. 2, pp. 302-312.

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Generalized Chebyshev polynomials are introduced and studied in this paper. They are applied to obtain a lower bound for the $\sup$-norm on the closed interval for nonzero polynomials with integer coefficients of arbitrary degree.
Keywords: extremal properties of polynomials, Hilbert–Fekete theorem, integer algebraic numbers, asymptotic law of the distribution of primes, Eisenstein criterion for the irreducibility of polynomials. 
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R. M. Trigub. Chebyshev Polynomials and Integer Coefficients. Matematičeskie zametki, Tome 105 (2019) no. 2, pp. 302-312. http://geodesic.mathdoc.fr/item/MZM_2019_105_2_a10/

[1] G. M. Goluzin, Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Gostekhizdat, M.–L., 1952 | MR | Zbl

[2] T. J. Rivlin, Chebyshev Polynomials. From Approximation Theory to Algebra and Number Theory, John Wiley Sons, New York, 1990 | MR | Zbl

[3] P. B. Borwein, C. G. Pinner, I. E. Pinsker, “Monic integer Chebyshev problem”, Math. Comp., 72:244 (2003), 1901–1916 | DOI | MR | Zbl

[4] M. Fekete, “Über die verzeilung der wurzeln bei gewissen algebraischen gleichungen mit ganzzahligen koeffizienten”, Math. Z., 17 (1923), 228–249 | DOI | MR | Zbl

[5] H. Hewilt, H. S. Zuckerman, “Approximation by polynomials with integral coefficients, a reformulation of the Stone–Weierstrass theorem”, Duke Math. J., 26 (1959), 305–324 | MR

[6] M. Fekete, “Approximation par polynomes over conditions diophantions”, C. R. Acad. Sci. Paris, 239 (1954), 1337–1339 | MR | Zbl

[7] Y. Okada, “On approximate polynomials with integral coefficients only”, Tôhoku Math. J., 23 (1923), 26–35 | Zbl

[8] Dzh. V. S. Kassels, Vvedenie v teoriyu diofantovykh priblizhenii, IL, M., 1961 | MR

[9] R. M. Trigub, “Priblizhenie funktsii s diofantovymi usloviyami mnogochlenami s tselymi koeffitsientami”, Metricheskie voprosy teorii funktsii i otobrazhenii, Vyp. 2, Naukova dumka, K., 1971, 267–353

[10] F. Amoroso, “Sur le diamétre transfini entier d'un intervalle reel”, Ann. Inst. Fourier (Grenoble), 40:4 (1991), 885–911 | DOI | MR

[11] B. S. Kashin, “Ob algebraicheskikh mnogochlenakh s tselymi koeffitsientami, malo uklonyayuschikhsya ot nulya na otrezke”, Matem. zametki, 50:3 (1991), 58–67 | MR | Zbl

[12] R. M. Trigub, “O priblizhenii gladkikh funktsii i konstant mnogochlenami s tselymi i naturalnymi koeffitsientami”, Matem. zametki, 70:1 (2001), 123–136 | DOI | MR | Zbl

[13] A. O. Gelfond, Kommentarii k statyam: P. L. Chebyshev, Sobranie sochinenii. T. 1. Teoriya chisel, M.–L., 1944, 285–288

[14] D. S. Gorshkov, “Ob uklonenii ot nulya polinomami s tselymi ratsionalnymi koeffitsientami v promezhutke $[0,1]$”, Trudy III Vsesoyuznogo matematicheskogo s'ezda, T. 4, AN SSSR, M., 1959, 5–7

[15] H. I. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., Providence, RI, 1994 | MR | Zbl

[16] K. G. Hare, “Generalized Gorshkov–Wirsing polynomials and the integer Chebyshev problem”, Exp. Math., 20:2 (2011), 189–200 | DOI | MR | Zbl

[17] P. Borwein, T. Erdélyi, “The integer Chebyshev problem”, Math. Comp., 65:214 (1996), 661–681 | DOI | MR | Zbl

[18] V. V. Prasolov, Mnogochleny, MTsMMO, M., 2003

[19] I. E. Pritsker, “Small polynomials with integer coefficients”, J. Anal. Math., 96 (2005), 151–190 | DOI | MR | Zbl