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A Note On the Hadamard Kth Root of a Rational Function

Published online by Cambridge University Press:  09 April 2009

Roberts Rumely  and
A. J. Van der Poorten
Roberts Rumely
Affiliation:
Department of MathematicsUniversity of GeorgiaAthens, Georgia 30602, U.S.A.
A. J. Van der Poorten
Affiliation:
School of Mathematics and PhysicsMacquarie UniversityNorth Ryde, N.S.W. 2113, Australia
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Abstract

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Suppose the sequence of Taylor coefficients of a rational function f consists of kth powers of elements all belonging to some finitely generated extension field F of Q. Then it is a generalisation of a conjecture of Pisot that there is a rational function with Taylor coefficients term-by-term kth roots of those of f. The authors show that it suffices to prove the conjecture in the case that the field of definition is a number field and prove the conjecture in that case subject to the constraint that f has a dominant pole, that is, that there is a valuation with respect to which f has a unique pole either of maximal or of minimal absolute value.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 43 , Issue 3 , December 1987 , pp. 314 - 327
Copyright
Copyright © Australian Mathematical Society 1987

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