Applications of Computational Algebraic Geometry: American Mathematical Society Short Course, January 6-7, 1997, San Diego, CaliforniaThis book introduces readers to key ideas and applications of computational algebraic geometry. Beginning with the discovery of Gröbner bases and fueled by the advent of modern computers and the rediscovery of resultants, computational algebraic geometry has grown rapidly in importance. The fact that "crunching equations" is now as easy as "crunching numbers" has had a profound impact in recent years. At the same time, the mathematics used in computational algebraic geometry is unusually elegant and accessible, which makes the subject easy to learn and easy to apply. This book begins with an introduction to Gröbner bases and resultants, then discusses some of the more recent methods for solving systems of polynomial equations. A sampler of possible applications follows, including computer-aided geometric design, complex information systems, integer programming, and algebraic coding theory. The lectures in this book assume no previous acquaintance with the material. |
Contents
Introduction to Gröbner bases | 1 |
Introduction to resultants | 25 |
Numerical methods for solving polynomial equations | 41 |
Applications to computer aided geometric design | 67 |
Combinatorial homotopy of simplicial complexes and complex information systems | 91 |
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Common terms and phrases
A-homotopy algebraic curve algebraic geometry algebraic sets algorithm applications Aq(A base point basis g Bernstein Bézier curve binomial Buchberger's algorithm called codeword coefficients combinatorial components computer algebra consider corresponding cost vector cubic cyclic code decoding defined degree denote determinant dimensional Editor eigenvalue eigenvectors elements elimination theory encoding example fiber finite formula given Goppa codes graph Graver basis Gröbner fan Hence homotopy ideal implicit equation integer programs inverse kinematics IPA.cGb kerz(A leading terms LEMMA linear linear code LM(f LT(f Mathematics matrix polynomial methods minimal monomial order multiple nonzero number of solutions optimal solution paths in Ta(A plane polynomial equations polynomial systems polytope posz(A problem PROOF PROPOSITION q-chain q-connected rational reduced Gröbner basis Reed–Solomon codes respect robotics roots S-polynomial Section simplex simplicial complex singular solving sparse resultant St(A Sturmfels surface system of polynomial term order test set Theorem UGBA unique variables vertex zero