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• W3103295634 endingPage "2019" @default.
• W3103295634 startingPage "1997" @default.
• W3103295634 abstract "We study the problem of finding nonconstant monic integer polynomials, normalized by their degree, with small supremum on an interval <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding=application/x-tex>I</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The monic integer transfinite diameter <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t Subscript normal upper M Baseline left-parenthesis upper I right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>M</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>t_{mathrm {M}}(I)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is defined as the infimum of all such supremums. We show that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding=application/x-tex>I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has length <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding=application/x-tex>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t Subscript normal upper M Baseline left-parenthesis upper I right-parenthesis equals one half> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>M</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mstyle displaystyle=false scriptlevel=0> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> </mml:mrow> <mml:annotation encoding=application/x-tex>t_{mathrm {M}}(I) = tfrac {1}{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We make three general conjectures relating to the value of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t Subscript normal upper M Baseline left-parenthesis upper I right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>M</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>t_{mathrm {M}}(I)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for intervals <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding=application/x-tex>I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of length less than <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=4> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding=application/x-tex>4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also conjecture a value for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t Subscript normal upper M Baseline left-parenthesis left-bracket 0 comma b right-bracket right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>M</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy=false>]</mml:mo> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>t_{mathrm {M}}([0,b])</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=0 greater-than b less-than-or-equal-to 1> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>b</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>0>ble 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We give some partial results, as well as computational evidence, to support these conjectures. We define functions <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Subscript minus Baseline left-parenthesis t right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L_{-}(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Subscript plus Baseline left-parenthesis t right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>+</mml:mo> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L_{+}(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which measure properties of the lengths of intervals <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding=application/x-tex>I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t Subscript normal upper M Baseline left-parenthesis upper I right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>M</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>t_{mathrm {M}}(I)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on either side of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding=application/x-tex>t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Upper and lower bounds are given for these functions. We also consider the problem of determining <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t Subscript normal upper M Baseline left-parenthesis upper I right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>M</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>t_{mathrm {M}}(I)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding=application/x-tex>I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Farey interval. We prove that a conjecture of Borwein, Pinner and Pritsker concerning this value is true for an infinite family of Farey intervals." @default.
• W3103295634 created "2020-11-23" @default.
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• W3103295634 date "2006-06-16" @default.
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• W3103295634 title "The monic integer transfinite diameter" @default.
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