Résumé : I will start from reviewing Groebner bases and their connection to polynomial system solving. The problem of solving a polynomial system of equations over a finite field has relevant applications to cryptography and coding theory. For many of these applications, being able to estimate the complexity of computing a Groebner basis is crucial. With these applications in mind, I will review linear-algebra based algorithms, which are currently the most efficient algorithms available to compute Groebner bases. I will define and compare several invariants, that were introduced with the goal of providing an estimate on the complexity of computing a Groebner basis, including the solving degree, the degree of regularity, and the last fall degree. Concrete examples will complement the theoretical discussion.
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