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The integer Chebyshev problem. (English) Zbl 0859.11044
This paper gives a good indication of the great difficulties that can be expected if one wants to improve (even slightly) upon previous results for very easily formulated problems. Nevertheless, the authors have been able to strike a good balance between the description of methods and explicitly writing down proofs in this paper.
Consider the following minimization problem: \[ \Omega_n[a,b]: =\biggl(\inf\bigl\{ |p|_{[a,b]}: 0\neq p\in Z_n\bigr\} \biggr)^{1/n} \] \((Z_n\): polynomials of degree at most \(n\) with integer coefficients; \(|\cdot |\): the sup-norm). \(\Omega[a,b]: =\inf\{\Omega_n[a,b]: n=0,1,2,\dots\} =\lim_{n\to\infty} \Omega_n[a,b ]\). A polynomial for which the value \(\Omega_n[a,b]\) is attained is called an \(n\)-th integer Chebyshev polynomial.
Much is known (cf. the references in the paper) and as it is sufficient to restrict to intervals of length at most 4 (for \(b-a>4\): \(\Omega[a,b] = \Omega_n [a,b] =1)\), the authors look at the interval \([0,1]\). The best previous bounds are \(1/2.37684 \dots \leq \Omega [0,1]\leq 1/2.3541\dots\). The upper bound is improved to \(1/2.3605 \dots\) and the authors show that for every natural number \(k\), the \(n\)-th integer Chebyshev polynomial is divisible by \((P_{120})^k\), with \(P_{120}\) a fixed integer polynomial of degree 120, provided \(n\) is sufficiently large.
Using results on orthogonal Müntz-Legendre polynomials, the authors furthermore deduce other results on factors of \(n\)-th integer Chebyshev polynomials and the dependence of \(\Omega[a,b]\) on the interval \([a,b]\). To get an impression of the type of results, the following is proved: \[ \left(m+2- {1\over 4(m+1)} \right)^{-1} \leq\Omega [0,1/m] \text{ for every } m=1,2, \ldots, \]
\[ \Omega[0,1/m] \leq(m+1.46)^{-1} \text{ for } m \text{ large enough}. \] Finally the connection with the Schur-Siegel trace problem is indicated and it is shown how bounds on \(\Omega[0,1/m]\) can be used to give a lower bound for \((\alpha_1+ \cdots + \alpha_d)/d\), where \(\alpha_1\) is a totally positive algebraic integer of degree \(d\).

MSC:
11J54 Small fractional parts of polynomials and generalizations
11B83 Special sequences and polynomials
41A50 Best approximation, Chebyshev systems
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