Giac and GeoGebra – Improved Gröbner Basis Computations

Abstract

GeoGebra is open source mathematics education software being used in thousands of schools worldwide. It already supports equation system solving, locus equation computation and automatic geometry theorem proving by using an embedded or outsourced CAS. GeoGebra recently changed its embedded CAS from Reduce to Giac because it fits better into the educational use. Also careful benchmarking of open source Gröbner basis implementations showed that Giac is fast in algebraic computations, too, therefore it allows heavy Gröbner basis calculations even in a web browser via Javascript.

Gröbner basis on ${\mathbb {Q}}$ for revlex ordering implementation in Giac is a modular algorithm (E. Arnold). Each ${\mathbb {Z}}/p{\mathbb {Z}}$ computation is done via the Buchberger algorithm using F4 linear algebra technics and “remake” speedups, they might be run in parallel for large examples. The output can be probabilistic or certified (which is much slower). Experimentation shows that the probabilistic version is faster than other open-source implementations, and about 3 times slower than the Magma implementation on one processor, it also requires less memory for big examples like Cyclic9.