Fast algorithms for differential equations in positive characteristic

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 ˜O(p1/2 ), and for determining a whole basis of the solution space in quasi-linear time ˜O(p); the ˜O notation indicates that we hide logarithmic factors. We show that for equations of arbitrary order, the

• Type: Conference
• Year: 2009
• Where: ISSAC
• Authors: Alin Bostan, Éric Schost
• Keywords: Complexity Issues, ISSAC 2009, Linear Differential Equations, Mathematics, Polynomial Solutions