Hypergeometric dispersion and the orbit problem

Authors:
Sergei A. Abramov;Manuel Bronstein
Affiliations:
Computer Center of the Russian Academy of Science, Vavilova 40, Moscow 117967, Russia;INRIA - Projet Cafe, 2004, Route des Lucioles, B.P. 93, F-06902 Sophia Antipolis Cedex, France
Venue:
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
Year:
2000

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Abstract

We describe an algorithm for finding the positive integer solutions n of orbit problems of the form &agr;n = &bgr; where &agr; and &bgr; are given elements of a field K. Our algorithm corrects the bounds given in [7], and shows that the problem is not polynomial in the Euclidean norms of the polynomials involved. Combined with a simplified version of the algorithm of [8] for the “specification of equivalence”, this yields a complete algorithm for computing the dispersion of polynomials in nested hypergeometric extensions of rational function fields. This is a necessary step in computing symbolic sums, or solving difference equations, with coefficients in such fields. We also solve the related equations p(&agr;n) = 0 and p(n, &agr;n) = 0 where p is a given polynomial and &agr; is given.