Fast algorithms for differential equations in positive characteristic

  • Authors:

  • Alin Bostan;Éric Schost

  • Affiliations:

  • INRIA Paris-Rocquencourt, Le Chesnay, France;The University of Western Ontario, London, ON, Canada

  • Venue:
  • Proceedings of the 2009 international symposium on Symbolic and algebraic computation
  • Year:
  • 2009

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Abstract

We address complexity issues for linear differential equations in characteristic p ;0: resolution and computation of the p-curvature. For these tasks, our main focus is on algorithms whose complexity behaves well with respect to p. We prove bounds linear in p on the degree of polynomial solutions and propose algorithms for testing the existence of polynomial solutions in sublinear time Õ(p1/2), and for determining a whole basis of the solution space in quasi-linear time Õ(p); the Õ notation indicates that we hide logarithmic factors. We show that for equations of arbitrary order, the p-curvature can be computed in subquadratic time Õ(p1.79), and that this can be improved to O(log(p)) for first order equations and to Õ(p) for classes of second order equations.