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• W2000890953 abstract "We are concerned with the problem of minimizing the supremum norm on an interval of a nonzero polynomial of degree at most<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n><mml:semantics><mml:mi>n</mml:mi><mml:annotation encoding=application/x-tex>n</mml:annotation></mml:semantics></mml:math></inline-formula>with integer coefficients. This is an old and hard problem that cannot be exactly solved in any nontrivial cases. We examine the case of the interval<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-bracket 0 comma 1 right-bracket><mml:semantics><mml:mrow><mml:mo stretchy=false>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=false>]</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>[0,1]</mml:annotation></mml:semantics></mml:math></inline-formula>in most detail. Here we improve the known bounds a small but interesting amount. This allows us to garner further information about the structure of such minimal polynomials and their factors. This is primarily a (substantial) computational exercise. We also examine some of the structure of such minimal “integer Chebyshev” polynomials, showing for example that on small intevals<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-bracket 0 comma delta right-bracket><mml:semantics><mml:mrow><mml:mo stretchy=false>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>δ<!-- δ --></mml:mi><mml:mo stretchy=false>]</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>[0, delta ]</mml:annotation></mml:semantics></mml:math></inline-formula>and for small degrees<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d><mml:semantics><mml:mi>d</mml:mi><mml:annotation encoding=application/x-tex>d</mml:annotation></mml:semantics></mml:math></inline-formula>,<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x Superscript d><mml:semantics><mml:msup><mml:mi>x</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>d</mml:mi></mml:mrow></mml:msup><mml:annotation encoding=application/x-tex>x^{d}</mml:annotation></mml:semantics></mml:math></inline-formula>achieves the minimal norm. There is a natural conjecture, due to the Chudnovskys and others, as to what the “integer transfinite diameter” of<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-bracket 0 comma 1 right-bracket><mml:semantics><mml:mrow><mml:mo stretchy=false>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=false>]</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>[0,1]</mml:annotation></mml:semantics></mml:math></inline-formula>should be. We show that this conjecture is false. The problem is then related to a trace problem for totally positive algebraic integers due to Schur and Siegel. Several open problems are raised." @default.
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• W2000890953 title "The integer Chebyshev problem" @default.
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• W2000890953 doi "https://doi.org/10.1090/s0025-5718-96-00702-8" @default.
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