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- W2008522364 abstract "The maximal modulus of an algebraic integer is the absolute value of its largest conjugate. We compute the minimum of the maximal modulus of all algebraic integers of degree <italic>d</italic> which are not roots of unity, for <italic>d</italic> at most 12. The computations suggest that the minimum is never attained for a reciprocal algebraic integer. The truth of this conjecture would show that the conjecture of Schinzel and Zassenhaus follows from a theorem of Smyth. We further test our conjecture by computing the minimum of the maximal modulus of all reciprocal algebraic integers of degree <italic>d</italic> which are not roots of unity, for <italic>d</italic> at most 16. Our computations strongly suggest that the best constant in the conjecture of Schinzel and Zassenhaus is 1.5 <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=log theta 0> <mml:semantics> <mml:mrow> <mml:mi>log</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>θ<!-- θ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>log {theta _0}</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=theta 0> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>θ<!-- θ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{theta _0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the smallest P.V. number. They also shed some light on a recent conjecture of Lind concerning the Perron numbers." @default.
- W2008522364 created "2016-06-24" @default.
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- W2008522364 date "1985-01-01" @default.
- W2008522364 modified "2023-10-14" @default.
- W2008522364 title "The maximal modulus of an algebraic integer" @default.
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- W2008522364 doi "https://doi.org/10.1090/s0025-5718-1985-0790657-8" @default.
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