APPLICATIONS OF ALGEBRAIC GEOMETRY
CONTENTS INTRODUCTION POLYNOMIALS AND IDEALS MONOMIAL ORDERS AND POLYNOMIAL DIVISION GR¨OBNER BASES AF?NE VARIETIES S L I G P L N M A E U T O S SOLVING POLYNOMIAL SYSTEMS BY ELIMINATION FINITE-DIMENSIONAL ALGEBRAS GR¨OBNER BASIS CONVERSION SOLVING EQUATIONS VIA EIGENVALUES AND EIGENVECTORS REAL ROOT LOCATION AND ISOLATION RESULTANTS THE RESULTANT OF TWO POLYNOMIALS MULTIPOLYNOMIAL RESULTANTS PROPERTIES OF RESULTANTS COMPUTING RESULTANTS SOLVING EQUATIONS VIA RESULTANTS COMPUTATION IN LOCAL RINGS LOCAL RINGS MULTIPLICITIES AND MILNOR NUMBERS TERM ORDERS AND DIVISION IN LOCAL RINGS STANDARD BASES IN LOCAL RINGS APPLICATIONS OF STANDARD BASES MODULES MODULES OVER RINGS MONOMIAL ORDERS AND GR¨OBNER BASES FOR MODULES COMPUTING SYZYGIES MODULES OVER LOCAL RINGS F EE R SOLUT O S PRESENTATIONS AND RESOLUTIONS OF MODULES HILBERT’S SYZYGY THEOREM GRADED RESOLUTIONS HILBERT POLYNOMIALS AND GEOMETRIC APPLICATIONS POLYTOPES, RESULTANTS, AND EQUATIONS GEOMETRY OF POLYTOPES SPARSE RESULTANTS TORIC VARIETIES MINKOWSKI SUMS AND MIXED VOLUMES BERNSTEIN’S THEOREM COMPUTING RESULTANTS AND SOLVING EQUATIONS POLYHEDRAL REGIONS AND POLYNOMIALS INTEGER PROGRAMMING INTEGER PROGRAMMING AND COMBINATORICS MULTIVARIATE POLYNOMIAL SPLINES THE GR¨OBNER FAN OF AN IDEAL THE GR¨OBNER WALK ALGEBRAIC CODING THEORY FINITE FIELDS ERROR-CORRECTING CODES CYCLIC CODES REED-SOLOMON DECODING ALGORITHMS 10 THE BERLEKAMP-MASSEY-SAKATA DECODING ALGORITHM CODES FROM ORDER DOMAINS THE OVERALL STRUCTURE OF THE BMS ALGORITHM THE DETAILS OF THE BMS ALGORITHM REFERENCES CHAPTER 2 X?N N 2X4 IS A MONOMIAL OF TOTAL DEGREE 6 IN THE VARIABLES X1, X2, X3, X4, SINCE ? = (3, 2, 0, 1) AND |?| = 6 IF K IS ANY ?ELD, WE CAN FORM ?NITE LINEAR COMBINATIONS OF MONOMIALS WITH COE?CIENTS IN K THE RESULTING OBJECTS ARE KNOWN AS POLYNOMIALS IN X1, , XN WE WILL ALSO USE THE WORD TERM ON OCCASION TO REFER TO A PRODUCT OF A NONZERO ELEMENT OF K AND A MONOMIAL APPEARING IN A POLYNOMIAL THUS, A GENERAL POLYNOMIAL IN THE VARIABLES X1, , XN WITH COE?CIENTS IN K HAS THE FORM F = ? C?X?, WHERE C? ?K FOR EACH ?, AND THERE ARE ONLY ?NITELY MANY TERMS C?X? IN THE SUM FOR EXAMPLE, TAKING K TO BE THE ?ELD Q OF RATIONAL NUMBERS, AND DENOTING THE VARIABLES BY X, Y, Z RATHER THAN USING SUBSCRIPTS, (1 2) P = X2 + IS A POLYNOMIAL CONTAINING FOUR TERMS IN MOST OF OUR EXAMPLES, THE ?ELD OF COE?CIENTS WILL BE EITHER Q, THE ?ELD OF REAL NUMBERS, R, OR THE ?ELD OF COMPLEX NUMBERS, C POLYNOMI- ALS OVER ?NITE ?ELDS WILL ALSO BE INTRODUCED IN CHAPTER 9 WE WILL DENOTE BY K[X1, , XN] THE COLLECTION OF ALL POLYNOMIALS IN X1, , XN WITH CO- E?CIENTS IN K POLYNOMIALS IN K[X1, , XN] CAN BE ADDED AND MULTIPLIED AS USUAL, SO K[X1, , XN] HAS THE STRUCTURE OF A COMMUTATIVE RING (WITH IDENTITY) HOWEVER, ONLY NONZERO CONSTANT POLYNOMIALS HAVE MULTIPLICATIVE INVERSES IN K[X1, , XN], SO K[X1, , XN] IS NOT A ?ELD HOWEVER, THE SET OF RATIONAL FUNCTIONS {F/G : F, G ?K[X1, , XN], G ?= 0} IS A ?ELD, DENOTED K(X1, , XN) A POLYNOMIAL F IS SAID TO BE HOMOGENEOUS IF ALL THE MONOMIALS APPEARING IN IT WITH NONZERO COE?CIENTS HAVE THE SAME TOTAL DEGREE FOR INSTANCE, F = 4X3 + 5XY2 ?Z3 IS A HOMOGENEOUS POLYNOMIAL OF TOTAL DEGREE 3 IN Q[X, Y, Z], WHILE G = 4X3 + 5XY2 ?Z6 IS NOT HOMOGENEOUS WHEN WE STUDY RESULTANTS IN CHAPTER 3, HOMOGENEOUS POLYNOMIALS WILL PLAY AN IMPORTANT ROLE GIVEN A COLLECTION OF POLYNOMIALS, F1, , FS ?K[X1, , XN], WE CAN CONSIDER ALL POLYNOMIALS WHICH CAN BE BUILT UP FROM THESE BY MULTIPLICATION BY ARBITRARY POLYNOMIALS AND BY TAKING SUMS (1 3) DE?NITION LET F1, , FS ?K[X1, , XN] WE LET ?F1, , FS? DENOTE THE COLLECTION ?F1, , FS?= {P1F1 + + PSFS : PI ?K[X1, , XN] FOR I = 1, , S} §1 POLYNOMIALS AND IDEALS F2 = X2 + Y2 + (Z ?1)2 ?4 WE HAVE (1 4) P = X2 + 2 Z + 1)(X2 + Z2 ?1) + (