On the complexity of computing with zero-dimensional triangular sets

We study the complexity of some fundamental operations for triangular sets in dimension zero. Using Las-Vegas algorithms, we prove that one can perform such operations as change of order, equiprojectable decomposition, or quasi-inverse computation with a cost that is essentially that of modular composition. Over an abstract field, this leads to a subquadratic cost (with respect to the degree of the underlying algebraic set). Over a finite field, in a boolean RAM model, we obtain a quasi-linear running time using Kedlaya and Umans' algorithm for modular composition. Conversely, we also show how to reduce the problem of modular composition to change of order for triangular sets, so that all these problems are essentially equivalent. Our algorithms are implemented in Maple; we present some experimental results

Fast Arithmetics in Artin-Schreier Towers over Finite Fields

An Artin-Schreier tower over the finite field F_p is a tower of field extensions generated by polynomials of the form X^p - X - a. Following Cantor and Couveignes, we give algorithms with quasi-linear time complexity for arithmetic operations in such towers. As an application, we present an implementation of Couveignes' algorithm for computing isogenies between elliptic curves using the p-torsion.Comment: 28 pages, 4 figures, 3 tables, uses mathdots.sty, yjsco.sty Submitted to J. Symb. Compu

Fast Conversion Algorithms for Orthogonal Polynomials

We discuss efficient conversion algorithms for orthogonal polynomials. We describe a known conversion algorithm from an arbitrary orthogonal basis to the monomial basis, and deduce a new algorithm of the same complexity for the converse operation

Algorithms for the universal decomposition algebra

Let k be a field and let f be a polynomial of degree n in k [T]. The symmetric relations are the polynomials in k [X1, ..., Xn] that vanish on all permutations of the roots of f in the algebraic closure of k. These relations form an ideal Is; the universal decomposition algebra is the quotient algebra A := k [X1, ..., Xn]/Is. We show how to obtain efficient algorithms to compute in A. We use a univariate representation of A, i.e. an explicit isomorphism of the form A=k [T]/Q (T), since in this representation, arithmetic operations in A are known to be quasi-optimal. We give details for two related algorithms, to find the isomorphism above, and to compute the characteristic polynomial of any element of A

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