Maple-based introductory visual guide to Gröbner bases

In 1975 the Consejo Superior de Investigaciones Científicas (the main Spanish institution for scientific research) published the monograph [14] by the second author (by the way, father and Ph.D. advisor of the first author). Its title could be translated as "Geometric Interpretation of Ideal Theory" (nowadays Ideal Theory is not normally used, in favour of Commutative Algebra). It somehow illustrated the geometric ideas underlying the basics of the classic books of the period (like [2, 11, 16]) and was a success: although written in Spanish, the edition was sold out.Of course there are much more modern books on ideals and varieties than [2, 11, 16], such as the famous [7] or [8], that illustrate the theory with images. Moreover, there are introductory works to Gröbner bases such as [3, 9, 12, 13, 15], as well as books on the topic like [1], and articles about applications, like the early [4]. Even a summary in English of the original Ph.D. Thesis by Bruno Buchberger is available [5]. Nevertheless, we believe that there is a place for a visual guide to Gröbner bases, as there was a place for [14]. For instance, statistical packages are probably the pieces of mathematical software best known by non-mathematicians, and they are frequently used as black boxes by users with a slight knowledge of the theory behind. Meanwhile, Gröbner bases, the most common exact method behind non-linear polynomial systems (algebraic systems) solving, although incorporated to all computer algebra systems, are only known by a relatively small ratio of the members of the scientific community, most of them mathematicians. This article presents in an intuitive and visual way an illustrative selection of ideals and their Gröbner bases, together with the plots of the (real part) of their corresponding algebraic varieties, computed and plotted with Maple [6, 10]. A minimum amount of theoretical details is given. We believe that exact algebraic systems solving could also be used as a black box by non-mathematicians just understanding the basic ideas underlying commutative algebra and computer algebra.