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W2068796687 abstract "Let<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$nge 2$ id=E1><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>be an integer and let<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$P={1,2,dotsc,n, n+1}$ id=E2><mml:mi>P</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math>. Let<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$Z_p$ id=E3><mml:msub><mml:mi>Z</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math>denote the finite field<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=${0,1,2,dotsc,p-1}$ id=E4><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math>, where<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$p ge 2$ id=E5><mml:mi>p</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:math>is a prime. Then every map<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$sigma$ id=E6><mml:mi>σ</mml:mi></mml:math>on<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$P$ id=E7><mml:mi>P</mml:mi></mml:math>determines a real<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$ntimes n$ id=E8><mml:mi>n</mml:mi><mml:mo>×</mml:mo><mml:mi>n</mml:mi></mml:math>Petrie matrix<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$A_{sigma}$ id=E9><mml:msub><mml:mi>A</mml:mi><mml:mi>σ</mml:mi></mml:msub></mml:math>which is known to contain information on the dynamical properties such as topological entropy and the Artin-Mazur zeta function of the linearization of<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$sigma$ id=E10><mml:mi>σ</mml:mi></mml:math>. In this paper, we show that if<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$sigma$ id=E11><mml:mi>σ</mml:mi></mml:math>is a cyclic permutation on<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$P$ id=E12><mml:mi>P</mml:mi></mml:math>, then all such matrices<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$A_{sigma}$ id=E13><mml:msub><mml:mi>A</mml:mi><mml:mi>σ</mml:mi></mml:msub></mml:math>are similar to one another over<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$Z_2$ id=E14><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>(but not over<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$Z_p$ id=E15><mml:msub><mml:mi>Z</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math>for any prime<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$pge 3$ id=E16><mml:mi>p</mml:mi><mml:mo>≥</mml:mo><mml:mn>3</mml:mn></mml:math>) and their characteristic polynomials over<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$Z_2$ id=E17><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>are all equal to<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$sum_{k=0}^n x^k$ id=E18><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:msubsup><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:math>. As a consequence, we obtain that if<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$sigma$ id=E19><mml:mi>σ</mml:mi></mml:math>is a cyclic permutation on<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$P$ id=E20><mml:mi>P</mml:mi></mml:math>, then the coefficients of the characteristic polynomial of<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$A_{sigma}$ id=E21><mml:msub><mml:mi>A</mml:mi><mml:mi>σ</mml:mi></mml:msub></mml:math>are all odd integers and hence nonzero." @default.
W2068796687 created "2016-06-24" @default.
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W2068796687 date "2004-01-01" @default.
W2068796687 modified "2023-09-25" @default.
W2068796687 title "On the class of square Petrie matrices induced by cyclic permutations" @default.
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