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Computational Commutative Algebra 2 is the natural continuation of Computational Commutative Algebra 1 with some twists, starting with the differently coloured cover graphics. The first volume had 3 chapters, 20 sections, 44 tutorials, and some amusing quotes. Since bigger is better, this book contains 3 chapters filling almost twice as many pages, 23 sections (some as big as a whole chapter), and 55 tutorials (some as big as a whole section). The number of jokes and quotes has increased exponentially due to the little-known fact that a good mathematical joke is better than a dozen mediocre papers. The main part of this book is a breathtaking passeggiata through the computational domains of graded rings and modules and their Hilbert functions. Besides Gröbner bases, we encounter Hilbert bases, border bases, SAGBI bases, and even SuperG bases. The tutorials traverse areas ranging from algebraic geometry and combinatorics to photogrammetry, magic squares, coding theory, statistics, and automatic theorem proving. Whereas in the first volume gardening and chess playing were not treated, in this volume they are. This is a book for learning, teaching, reading, and most of all, enjoying the topic at hand. The theories it describes can be applied to anything from children's toys to oil production. If you buy it, probably one spot on your desk will be lost forever!

Gröbner bases --- Commutative algebra --- Grobner, Bases de --- Algebre commutative --- Data processing --- Informatique --- Commutative algebra -- Data processing. --- Gröbner bases. --- Gröbner basis theory --- Mathematics. --- Computer science --- Algebra. --- Algebraic geometry. --- Computer mathematics. --- Algorithms. --- Computational Mathematics and Numerical Analysis. --- Symbolic and Algebraic Manipulation. --- Algebraic Geometry. --- Data processing. --- Algebra --- Geometry, algebraic. --- Algebraic geometry --- Geometry --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Algorism --- Arithmetic --- Mathematics --- Mathematical analysis --- Foundations --- Combinatorial analysis. --- QA 150-272 Algebra. --- Computer science—Mathematics. --- Group theory. --- Group Theory and Generalizations. --- Groups, Theory of --- Substitutions (Mathematics) --- Commutative algebra - Data processing --- Algebre commutative - Informatique

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Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated? The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory. The algorithms to answer questions such as those posed above are an important part of algebraic geometry. This book bases its discussion of algorithms on a generalization of the division algorithm for polynomials in one variable that was only discovered in the 1960's. Although the algorithmic roots of algebraic geometry are old, the computational aspects were neglected earlier in this century. This has changed in recent years, and new algorithms, coupled with the power of fast computers, have let to some interesting applications, for example in robotics and in geometric theorem proving. In preparing a new edition of Ideals, Varieties and Algorithms the authors present an improved proof of the Buchberger Criterion as well as a proof of Bezout's Theorem. Appendix C contains a new section on Axiom and an update about Maple , Mathematica and REDUCE.

Geometry, Algebraic --- Commutative algebra --- Data processing --- Algebraic geometry --- Algèbres commutatives --- Géométrie algébrique --- Algèbres commutatives --- Géométrie algébrique --- Computer. Automation --- Informatique --- Mathematical logic. --- Mathematical Logic and Foundations. --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Geometry, Algebraic - Data processing --- Commutative algebra - Data processing --- Géométrie algébrique. --- Algorithms --- Algorithmes. --- Algèbres commutatives.