COMPUTATIONAL COMMUTATIVE AND NON-COMMUTATIVE ALGEBRAIC GEOMETRY
CONTENTS CHAPTER 1 GEOMETRY ALGEBRA AND ALGORITHMS 1 1 POLYNOMIALS AND AF NE SPACE 1 2 AF NE VARIETIES 5 3 PARAMETRIZATIONS OF AF NE VARIETIES 14 4 IDEALS 29 5 POLYNOMIALS OF ONE VARIABLE 37 CHAPTER 2 GR BNER BASES 49 1 INTRODUCTION 49 2 ORDERINGS ON THE MONOMIALS IN K X1 XN 54 3 A DIVISION ALGORITHM IN K X1 XN 61 4 MONOMIAL IDEALS AND DICKSON S LEMMA 70 5 THE HILBERT BASIS THEOREM AND GR BNER BASES 76 6 PROPERTIES OF GR BNER BASES 83 7 BUCHBERGER S ALGORITHM 90 8 FIRST APPLICATIONS OF GR BNER BASES 97 9 RE NEMENTS OF THE BUCHBERGER CRITERION 104 10 IMPROVEMENTS ON BUCHBERGER S ALGORITHM 109 CHAPTER 3 ELIMINATION THEORY 121 1 THE ELIMINATION AND EXTENSION THEOREMS 121 2 THE GEOMETRY OF ELIMINATION 129 3 IMPLICITIZATION 133 4 SINGULAR POINTS AND ENVELOPES 143 5 GR BNER BASES AND THE EXTENSION THEOREM 155 6 RESULTANTS AND THE EXTENSION THEOREM 161 CHAPTER 4 THE ALGEBRA GEOMETRY DICTIONARY 175 1 HILBERT S NULLSTELLENSATZ 175 2 RADICAL IDEALS AND THE IDEAL VARIETY CORRESPONDENCE 181 3 SUMS PRODUCTS AND INTERSECTIONS OF IDEALS 189 4 ZARISKI CLOSURES IDEAL QUOTIENTS AND SATURATIONS 199 5 IRREDUCIBLE VARIETIES AND PRIME IDEALS 206 6 DECOMPOSITION OF A VARIETY INTO IRREDUCIBLES 212 7 PROOF OF THE CLOSURE THEOREM 219 8 PRIMARY DECOMPOSITION OF IDEALS 228 9 SUMMARY 232 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS ON A VARIETY 233 1 POLYNOMIAL MAPPINGS 233 2 QUOTIENTS OF POLYNOMIAL RINGS 240 3 ALGORITHMIC COMPUTATIONS IN K X1 XN I 248 4 THE COORDINATE RING OF AN AF NE VARIETY 257 5 RATIONAL FUNCTIONS ON A VARIETY 268 6 RELATIVE FINITENESS AND NOETHER NORMALIZATION 277 CHAPTER 6 ROBOTICS AND AUTOMATIC GEOMETRIC THEOREM PROVING 291 1 GEOMETRIC DESCRIPTION OF ROBOTS 291 2 THE FORWARD KINEMATIC PROBLEM 297 3 THE INVERSE KINEMATIC PROBLEM AND MOTION PLANNING 304 4 AUTOMATIC GEOMETRIC THEOREM PROVING 319 5 WU S METHOD 335 CHAPTER 7 INVARIANT THEORY OF FINITE GROUPS 345 1 SYMMETRIC POLYNOMIALS 345 2 FINITE MATRIX GROUPS AND RINGS OF INVARIANTS 355 3 GENERATORS FOR THE RING OF INVARIANTS 364 4 RELATIONS AMONG GENERATORS AND THE GEOMETRY OF ORBITS 373 CHAPTER 8 PROJECTIVE ALGEBRAIC GEOMETRY 385 1 THE PROJECTIVE PLANE 385 2 PROJECTIVE SPACE AND PROJECTIVE VARIETIES 396 3 THE PROJECTIVE ALGEBRA GEOMETRY DICTIONARY 406 4 THE PROJECTIVE CLOSURE OF AN AF NE VARIETY 415 5 PROJECTIVE ELIMINATION THEORY 422 6 THE GEOMETRY OF QUADRIC HYPERSURFACES 436 7 BEZOUT S THEOREM 451 CHAPTER 9 THE DIMENSION OF A VARIETY 469 1 THE VARIETY OF A MONOMIAL IDEAL 469 2 THE COMPLEMENT OF A MONOMIAL IDEAL 473 3 THE HILBERT FUNCTION AND THE DIMENSION OF A VARIETY 486 4 ELEMENTARY PROPERTIES OF DIMENSION 498 5 DIMENSION AND ALGEBRAIC INDEPENDENCE 506 6 DIMENSION AND NONSINGULARITY 515 7 THE TANGENT CONE 525 CHAPTER 10 ADDITIONAL GR BNER BASIS ALGORITHMS 539 1 PRELIMINARIES 539 2 HILBERT DRIVEN BUCHBERGER ALGORITHMS 550 3 THE F4 ALGORITHM 567 4 SIGNATURE BASED ALGORITHMS AND F5 576 CHAPTER 1 GEOMETRY ALGEBRA AND ALGORITHMS THIS CHAPTER WILL INTRODUCE SOME OF THE BASIC THEMES OF THE BOOK THE GEOMETRY WE ARE INTERESTED IN CONCERNS AF NE VARIETIES WHICH ARE CURVES AND SURFACES AND HIGHER DIMENSIONAL OBJECTS DE NED BY POLYNOMIAL EQUATIONS TO UNDERSTAND AF NE VARIETIES WE WILL NEED SOME ALGEBRA AND IN PARTICULAR WE WILL NEED TO STUDY IDEALS IN THE POLYNOMIAL RING K X1 XN FINALLY WE WILL DISCUSS POLYNOMIALS IN ONE VARIABLE TO ILLUSTRATE THE ROLE PLAYED BY ALGORITHMS 1 POLYNOMIALS AND AF NE SPACE TO LINK ALGEBRA AND GEOMETRY WE WILL STUDY POLYNOMIALS OVER A ELD WE ALL KNOW WHAT POLYNOMIALS ARE BUT THE TERM ELD MAY BE UNFAMILIAR THE BASIC INTUITION IS THAT A ELD IS A SET WHERE ONE CAN DE NE ADDITION SUBTRACTION MULTIPLICATION AND DIVISION WITH THE USUAL PROPERTIES STANDARD EXAMPLES ARE THE REAL NUMBERS R AND THE COMPLEX NUMBERS C WHEREAS THE INTEGERS Z ARE NOT A ELD SINCE DIVISION FAILS 3 AND 2 ARE INTEGERS BUT THEIR QUOTIENT 3 2 IS NOT A FORMAL DE NITION OF ELD MAY BE FOUND IN APPENDIX A ONE REASON THAT ELDS ARE IMPORTANT IS THAT LINEAR ALGEBRA WORKS OVER ANY ELD THUS EVEN IF YOUR LINEAR ALGEBRA COURSE RESTRICTED THE SCALARS TO LIE IN R OR C MOST OF THE THEOREMS AND TECHNIQUES YOU LEARNED APPLY TO AN ARBITRARY ELD K IN THIS BOOK WE WILL EMPLOY DIFFERENT ELDS FOR DIFFERENT PURPOSES THE MOST COMMONLY USED ELDS WILL BE THE RATIONAL NUMBERS Q THE ELD FOR MOST OF OUR COMPUTER EXAMPLES THE REAL NUMBERS R THE ELD FOR DRAWING PICTURES OF CURVES AND SURFACES THE COMPLEX NUMBERS C THE ELD FOR PROVING MANY OF OUR THEOREMS ON OCCASION WE WILL ENCOUNTER OTHER ELDS SUCH AS ELDS OF RATIONAL FUNCTIONS WHICH WILL BE DE NED LATER THERE IS ALSO A VERY INTERESTING THEORY OF NITE ELDS SEE THE EXERCISES FOR ONE OF THE SIMPLER EXAMPLES 1