CHAPTER 1 Grobner Basic s Let k b e any field and k[x\ = k[x\,..., x n ] th e polynomia l rin g in n indetermi - nates. Th e monomial s i n fc[x] are denote d x a = Xi X 9 XyJ1 an d identifie d wit h lattice points a = (ai,... , an) i n N n , wher e N stand s for the non-negative integers . A tota l orde r - on N n i s a term order if th e zer o vecto r 0 i s th e uniqu e minima l element, an d a - b implie s a -f - c - b + c fo r al l a , b, c G Nn . Familia r example s of term order s are the purely lexicographic order, the degree lexicographic order and the degree reverse lexicographic order. Given a term orde r - , every non-zer o polynomia l / G k[x) ha s a unique initial monomial, denote d in^(f). I f I i s a n idea l i n fc[x], the n it s initial ideal i s th e monomial idea l in{I) : = (inM) •• / / ) . The monomial s whic h d o not li e in in^(I) ar e calle d standard monomials. A finite subset Q C / i s a Grobner basis for I wit h respec t t o - i f in^(I) i s generated b y {m^(g) : g G 6}. I f n o monomia l i n thi s se t i s redundant, the n th e Grobne r basi s Q is minimal. I t i s called reduced if, fo r an y two distinct element s g,g' G Q, no ter m of ( / i s divisibl e b y in^(g). Th e reduce d Grobne r basi s i s unique fo r a n idea l an d a ter m order , provide d on e require s th e coefficien t o f in^(g) i n g to b e 1 for eac h g G Q. Startin g with any set of generators for / , th e Buchberger algorithm compute s the reduce d Grobne r basi s Q. Th e division algorithm rewrite s eac h polynomia l / modulo / uniquel y a s a fc-linear combination o f standard monomials . Proposition 1.1 . The (images of the) standard monomials form a k-vector space basis for the residue ring k[x]/I. Clearly, ther e ar e infinitel y man y ter m order s i f n 2. However , i f th e idea l I i s fixed, the n the y ca n b e groupe d int o finitely man y equivalenc e classe s b y th e following theorem . Theorem 1.2 . Every ideal I C fc[x] has only finitely many distinct initial ideals. Proof: Suppos e tha t / ha s a n infinit e se t S o o f distinc t initia l ideals . Choos e a non-zero elemen t / i G /. Sinc e f\ ha s onl y finitely man y term s an d sinc e eac h term lie s i n a n elemen t o f EQ , ther e exist s a monomia l m\ i n f\ suc h tha t th e se t Ei : = { M G So : m\ G M} i s infinite . Sinc e (mi ) i s strictl y containe d i n a n initial idea l of /, Propositio n 1. 1 tell s us that th e monomial s outsid e o f (mi ) ar e £ - linearly dependent modul o /. Henc e there exists a non-zero polynomial J2^f non e of whos e term s lie s i n (mi) . Sinc e f^ ha s onl y finitely man y terms , ther e exist s a monomial m^ i n j^ such that th e set E 2 : = { M G Ei : 771 2 G M} i s infinite. Sinc e (mi,7712) is strictly containe d i n a n initia l idea l o f / , Propositio n 1. 1 tell s u s tha t the monomial s outside of (mi , 777,2) are /c-linearly dependent modul o / . Henc e ther e http://dx.doi.org/10.1090/ulect/008/01
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