Fast algorithms for differential equations in positive characteristic

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 timeÕ(p 1/2 ), and for determining a whole basis of the solution space in quasi-linear timeÕ(p); theÕ notation indicates that we hide logarithmic factors. We show that for equations of arbitrary order, the p-curvature can be computed in subquadratic timeÕ(p 1.79 ), and that this can be improved to O(log(p)) for first order equations and toÕ(p) for classes of second order equations.

Power series solutions of singular (q)-differential equations

“…Other motivations for considering algorithms that work in positive characteristic come from applications in numbertheory based cryptology or in combinatorics [7,8,10]. Our objectives in this respect are to overcome the lack of a Cauchy theorem, or of a formal theory of singular equations, by giving conditions that ensure the existence of solutions at the required precisions.…”

Algorithms for Structured Linear Systems Solving and Their Implementation

“…However, we can reduce this cost using babysteps / giant steps techniques: the bivariate modular composition algorithm of [36] shows that we can do matrix-vector product by T using O(e 1/2 M (de) + e (ω+1)/2 M (d)) operations; since multiplications by V u and W v take O(M (de)), we obtain the improved estimate C A = O(e 1/2 M (de) + e (ω+1)/2 M (d)) in this case. Algorithm DAC Q is the only algorithm we know of that takes into account the extra structure of algebraic approximants; we expect that similar improvements are possible for di erential approximants, using the evaluation algorithm of [12].…”

Enumeration and asymptotics of restricted compositions having the same number of parts

“…the smallest i such that (d/dz) i = L mod p is i = 3 for primes p = 2, 3, 5, 7, 11... and L is the above irreducible unreadable linear differential operator cancelling D(z)). Therefore, according to a conjecture of Grothendieck on the p-curvature (see [7]), the function is not algebraic. For m = 5, D(z) is a non algebraic function satisfying a differential equation of order 6 and of degree 38, which leads to…”