This introduction to polynomial rings, Gröbner bases and applications bridges the gap in the literature between theory and actual computation. It details numerous applications, covering fields as disparate as algebraic geometry and financial markets. To aid in a full understanding of these applications, more than 40 tutorials illustrate how the theory can be used. The book also includes many exercises, both theoretical and practical.
Hofstadter’s Law states that tasks always take longer than expected, even when you account for it. This second volume took four years to complete, confirming our writing velocity at one centimeter per year. When we began this project, our goal was to create a book that is not only educational but also enjoyable. We believe that a mathematical book doesn’t have to be dull or tedious. To achieve this, we incorporated amusing quotes, jokes, word games, and literary elements. However, this approach has its drawbacks. Humor is subjective, and not every metaphor resonates with everyone. For example, while joking about politicians can be amusing, it raises the question of how they might react to our words. Additionally, using metaphors like a small boat sailing into the Brazilian sunset can present geographical inaccuracies. Lastly, writing humorously in a foreign language poses its own challenges. Despite these potential pitfalls, our aim remains to engage readers and make the learning experience enjoyable.
Designed for students across various scientific and technological fields, this book covers the essential concepts of Linear Algebra. It serves as a foundational resource, making it particularly beneficial for those studying statistics as well. The content is structured to facilitate understanding of this critical subject, ensuring students grasp its applications and significance in their respective disciplines.
This book combines, in a novel and general way, an extensive development of the theory of families of commuting matrices with applications to zero-dimensional commutative rings, primary decompositions and polynomial system solving. It integrates the Linear Algebra of the Third Millennium, developed exclusively here, with classical algorithmic and algebraic techniques. Even the experienced reader will be pleasantly surprised to discover new and unexpected aspects in a variety of subjects including eigenvalues and eigenspaces of linear maps, joint eigenspaces of commuting families of endomorphisms, multiplication maps of zero-dimensional affine algebras, computation of primary decompositions and maximal ideals, and solution of polynomial systems. This book completes a trilogy initiated by the uncharacteristically witty books Computational Commutative Algebra 1 and 2 by the same authors. The material treated here is not available in book form, and much of it is notavailable at all. The authors continue to present it in their lively and humorous style, interspersing core content with funny quotations and tongue-in-cheek explanations.
