Polynomial ring automorphisms, rational (w,σ)-canonical forms, and the assignment problem

  • Authors:

  • S. A. Abramov;M. Petkovek

  • Affiliations:

  • Russian Academy of Sciences, Dorodnicyn Computing Centre, Vavilova 40, 119991, Moscow GSP-1, Russia;Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia

  • Venue:
  • Journal of Symbolic Computation
  • Year:
  • 2010

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Abstract

We investigate the representations of a rational function R@?k(x) where k is a field of characteristic zero, in the form R=K@?@sS/S. Here K,S@?k(x), and @s is an automorphism of k(x) which maps k[x] onto k[x]. We show that the degrees of the numerator and denominator of K are simultaneously minimized iff K=r/s where r,s@?k[x] and r is coprime with @s^ns for all n@?Z. Assuming existence of algorithms for computing orbital decompositions of R@?k(x) and semi-periods of irreducible p@?k[x]@?k, we present an algorithm for minimizing w(degnum(S),degden(S)) among representations with minimal K, where w is any appropriate weight function. This algorithm is based on a reduction to the well-known assignment problem of combinatorial optimization. We show how to use these representations of rational functions to obtain succinct representations of @s-hypergeometric terms.