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ROOTS OF UNITY AS QUOTIENTS OF TWO ROOTS OF A POLYNOMIAL

Part of: Sequences and sets Algebraic number theory: global fields

Published online by Cambridge University Press:  10 August 2012

ARTŪRAS DUBICKAS
ARTŪRAS DUBICKAS*
Affiliation:
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania (email: [email protected])
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Abstract

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Let K be a number field. For f∈K[x], we give an upper bound on the least positive integer T=T(f) such that no quotient of two distinct Tth powers of roots of f is a root of unity. For each ε>0 and each f∈ℚ[x] of degree d≥d(ε) we prove that . In the opposite direction, we show that the constant 2cannot be replaced by a number smaller than 1 . These estimates are useful in the study of degenerate and nondegenerate linear recurrence sequences over a number field K.

Keywords

root of unitynumber fieldlinear recurrence sequence

MSC classification

Secondary: 11R04: Algebraic numbers; rings of algebraic integers 11B37: Recurrences
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 92 , Issue 2 , April 2012 , pp. 137 - 144
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

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