J Austral Math Soc Ser A 43 pp314--327, 1987.

A Note on the Hadamard kth Root of a Rational Function

Robert S. Rumely and A. J. van der Poorten

Abstract

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.

1980 AMS Subject Classification: 10A35

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Authors

Robert S. Rumely

Department of Mathematics, University of Georgia, Athens, Georgia 30602, U.S.A.

A. J. van der Poorten

School of Mathematics and Physics, Macquarie University, North Ryde, N.S.W. 2113, Australia.