Alin Bostan

Polynomial evaluation and interpolation on special sets of points

We give complexity estimates for the problems of evaluation and interpolation on various polynomial bases. We focus on the particular cases when the sample points form an arithmetic or a geometric sequence, and we discuss applications, respectively, to computations with linear differential operators and to polynomial matrix multiplication.

Tellegen's principle into practice

The transposition principle, also called Tellegens principle, is a set of transformation rules for linear programs. Yet, though well known, it is not used systematically, and few practical implementations rely on it. In this article, we propose explicit transposed versions of polynomial multiplication and division but also new faster algorithms for multipoint evaluation, interpolation and their transposes. We report on their implementation in Shoups NTL C++ library.

FAST ALGORITHMS FOR COMPUTING ISOGENIES BETWEEN ELLIPTIC CURVES

We survey algorithms for computing isogenies between elliptic curves defined over a field of characteristic either 0 or a large prime. We introduce a new algorithm that computes an isogeny of degree l (l different from the characteristic) in time quasi-linear with respect to l. This is based in particular on fast algorithms for power series expansion of the Weierstrass ℘-function and related functions.

The complete generating function for Gessel walks is algebraic

Gessel walks are lattice walks in the quarter plane

Complexity issues in bivariate polynomial factorization

\set N^2

Differential equations for algebraic functions

which start at the origin

Globally nilpotent differential operators and the square Ising model

(0,0)\in\set N^2

Fast computation of special resultants

and consist only of steps chosen from the set

Hermite reduction and creative telescoping for hyperexponential functions

\{\leftarrow,\swarrow,\nearrow,\to\}

Complexity of creative telescoping for bivariate rational functions

. We prove that if