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On integer Chebyshev polynomials. (English) Zbl 0911.11033
The study of \(d_n=\text{LCM} (1,2,\ldots,n)\) in number theory motivates a closer look at polynomials \(P_k\in{\mathbb Z}_k[X]\) (integer coefficients, degree \(\leq k\)) and constants \(C_k\) defined by \[ | | P_k| | =\min_{P\in{\mathbb Z}_k[X]} | | P| | _{\infty},\quad C_k=-\tfrac 1k\log | | P_k| | _{\infty} \] (\(| | \cdot| | _{\infty}\) the Chebyshev norm on the interval \([0,1]\)). The polynomials \(P_k\) are called integer Bernstein polynomials on \([0,1]\) or polynomials of minimal diophantic deviation from zero.
Using MAPLE and two theoretical lemmas, the authors give a table of factorizations of \(P_k\) for \(k\leq 70\). In the polynomial \(P_{70}\) a factor \[ \begin{split} A_8(x)=4921 X^{10}-24605 X^9+53804 X^8-67586 X^7+ 53866 X^6\\ - 28388 X^5+9995 X^4-2317 X^3+338 X^2-28 X+1\end{split} \] shows up, leading to an improvement on the lower bound of \(C=\lim_{k\rightarrow\infty} C_k\) given in the literature and to the solution of an open problem in P. Borwein and T. Erdélyi [The integer Chebyshev problem, Math. Comput. 65, 661-681 (1996; Zbl 0859.11044)].

MSC:
11J54 Small fractional parts of polynomials and generalizations
41A10 Approximation by polynomials
41-04 Software, source code, etc. for problems pertaining to approximations and expansions
11-04 Software, source code, etc. for problems pertaining to number theory
Software:
Maple
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References:
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