We study the cost of multiplication modulo triangular families of polynomials. Following previous work by Li, Moreno Maza and Schost, we propose an algorithm that relies on homotopy and fast evaluation-interpolation techniques. We obtain a quasi-linear time complexity for substantial families of examples, for which no such result was known before. Applications are given to notably addition of algebraic numbers in small characteristic.

triangular sets

Homotopy techniques for multiplication modulo triangular sets

One of the main successes of the computer algebra community in the last 30 years has been the discovery of algorithms, called modular methods, that allow to keep the swell of the intermediate expressions under control. Without these methods, many applications of computer algebra would not be possible and the impact of computer algebra in scientific computing would be severely limited. Amongst the computer algebra systems which have emerged in the 70’s and 80’s, Maple and its developers have played an essential role in this area. Another major advance in symbolic computation is the development of implementation techniques for asymptotically fast (FFT-based) polynomial arithmetic. Computer algebra systems and libraries initiated in the 90’s, such as Magma and NTL, have been key actors in this effort: they increased in a spectacular manner the range of problems solvable by computer algebra systems. In this report, we present modpn, a Maple library dedicated to fast arithmetic for multivariate polynomials. The main objective of modpn is to provide highly efficient routines for supporting the implementation of modular methods in Maple. We demonstrate in this work that modpn allows us to re-implement core operations in Maple bringing huge performance increases and offering to Maple the ability of solving problems which were previously out of reach.