Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative AlgebraSpringer Science & Business Media, 17 d’abr. 2013 - 514 pàgines We wrote this book to introduce undergraduates to some interesting ideas in algebraic geometry and commutative algebra. Until recently, these topics involved a lot of abstract mathematics and were only taught in graduate school. But in the 1960's, Buchberger and Hironaka discovered new algorithms for manipulating systems of polynomial equations. Fueled by the development of computers fast enough to run these algorithms, the last two decades have seen a minor revolution in commutative algebra. The ability to compute efficiently with polynomial equations has made it possible to investigate complicated examples that would be impossible to do by hand, and has changed the practice of much research in algebraic geometry. This has also enhanced the importance of the subject for computer scientists and engineers, who have begun to use these techniques in a whole range of problems. It is our belief that the growing importance of these computational techniques warrants their introduction into the undergraduate (and graduate) mathematics curricu lum. Many undergraduates enjoy the concrete, almost nineteenth century, flavor that a computational emphasis brings to the subject. At the same time, one can do some substantial mathematics, including the Hilbert Basis Theorem, Elimination Theory and the Nullstellensatz. The mathematical prerequisites of the book are modest: the students should have had a course in linear algebra and a course where they learned how to do proofs. Examples of the latter sort of course include discrete math and abstract algebra. |
Continguts
1 | |
7 | |
Groebner Bases | 50 |
9 Optional Improvements on Buchbergers Algorithm | 101 |
Elimination Theory | 113 |
5 Unique Factorization and Resultants | 147 |
6 Resultants and the Extension Theorem | 159 |
Robotics and Automatic Geometric Theorem Proving | 255 |
1 Symmetric Polynomials | 312 |
3 Generators for the Ring of Invariants | 325 |
4 Relations among Generators and the Geometry of Orbits | 333 |
Projective Algebraic Geometry | 345 |
2 Groups | 480 |
1 Maple | 489 |
Independent Projects | 495 |
503 | |
Frases i termes més freqüents
a₁ affine variety algebraically closed Chapter compute a Groebner computer algebra systems consider containing coordinates Corollary curve defined Definition determine dimension divides division algorithm elements elimination ideal elimination theory example Exercise Extension Theorem field follows function ƒ and g geometric give given grevlex Groebner basis h₁ Hence Hilbert Hilbert Basis Theorem Hint homogeneous homogeneous coordinates homogeneous ideal implies integer intersection irreducible isomorphic joint k[xo leading coefficient leading terms Lemma Let f lex order linear LT(g LT(I mapping monomial ideal monomial order multidegree multiple Note Nullstellensatz parametrization polynomial equations polynomial ƒ prime ideal problem projective variety proof of Theorem Proposition prove quotient remainder Res(f ring robot S-polynomials singular points subset subspace suppose syzygies tangent total degree twisted cubic unique V(I₁ V₁ vanishes variables vector x²y xy²