A variedade das álgebras de Jordan de dimensão 2 e 3 a partir de bases de Gröbner

Abstract

In this work, we will present Gröbner basis and an algorithm to obtain them, as well as results from Algebraic Geometry on affine algebraic varieties. The calculation of Gröbner bases will allow us to understand the process of analyzing the classification of the affine variety of 2 and 3 dimensional Jordan algebras, which are not associative. For this purpose, we start with the study of the division algorithm in the polynomial ring κ[x_1,...,x_n] over an arbitrary field κ and its main characteristics, through different monomial orderings, detailing its algorithmic implementation. Then, the process of building a Gröbner base for a polynomial ideal I ⊂ κ[x_1,...,x_n] is studied, which allows us to answer, among other questions, the problem of ideal membership, and also some examples of computing Gröbner bases using the highly useful computational tool SageMath. Soon after, the concepts of affine space A^n(κ), affine variety V ⊂ A^n(κ) and their properties will be studied, in particular dimension and decomposition into irreducible components. Finally, as an application of Gröbner bases, we will present the classification of Jor_2(κ) and Jor_3(κ) over an algebraically closed field κ, studying its dimensions and irreducible components.