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. 
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/
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[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