Gröbner bases are the single most important tool in applicable algebraic geometry. They are used to compute standard representatives in the residue classes of a polynomial ring modulo an ideal and they can be used as a step towards solving a system of polynomial equations. Applications in science and technology are abundant, particularly in cryptography and coding theory.
Computation of Gröbner bases is challenging. It has computational complexity double exponential in the worst-case and exponential in average. Although this makes Gröbner bases intractable in many cases, a great deal of effort has been devoted to improve algorithms to compute faster larger Gröbner bases.
The concept of mutant polynomials, introduced by Ding in 2006, quantifies the deviation from the average case, leading to faster algorithms that profit on this degeneration.
In this dissertation we introduce three algorithms aiming at improving Gröbner bases computation that are inspired by mutant polynomials. The mutantF4 algorithm modifies Faugère's F4 by exploiting the presence of mutant polynomials. The MXL3 algorithm introduces a termination condition based on the absence of mutant polynomials. And the MGB algorithm combines MXL3 with the idea of preempted reduction to avoid storing large sets of polynomials. Each new idea achieved gains in speed and/or memory usage. The MGB algorithm is particularly successful, which was demonstrated by being the first algorithm to be able to compute a Gröbner basis for a random system of 32 quadratic polynomials in 32 variables over GF(2).
We also propose LASyz, a method to avoid redundant computation in Gröbner bases computation, that is compatible with mutant algorithms. LASyz is a simple yet effective method to significantly reduce unproductive reductions to zero. It uses linear algebra to maintain a set of generators for the module of syzygies. LASyz is compatible with mutant Gröbner basis algorithms thanks to its simplicity and the use of linear algebra for reducing both polynomials and syzygies.
Finally, we establish some theoretical results for mutant polynomials. We define the concept of mutant space, and study its dimension. We demonstrate that the mutant space is trivial in generic situations. We link the concept of mutants to the notions of regular sequences and syzygies. And we conjecture that the dimension of the mutant space constitutes a generic property of sequences of polynomials.