1. Notations and Definitions

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1 1. Notations and Definitions Objekttyp: Chapter Zeitschrift: L'Enseignement Mathématique Band (Jahr): 34 (1988) Heft 1-2: L'ENSEIGNEMENT MATHÉMATIQUE PDF erstellt am: Nutzungsbedingungen Die ETH-Bibliothek ist Anbieterin der digitalisierten Zeitschriften. Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften. Die Rechte liegen in der Regel bei den Herausgebern. Die auf der Plattform e-periodica veröffentlichten Dokumente stehen für nicht-kommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden. Das Veröffentlichen von Bildern in Print- und Online-Publikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt. Die systematische Speicherung von Teilen des elektronischen Angebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber. Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit. Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem Online-Angebot oder durch das Fehlen von Informationen. Dies gilt auch für Inhalte Dritter, die über dieses Angebot zugänglich sind. Ein Dienst der ETH-Bibliothek ETH Zürich, Rämistrasse 101, 8092 Zürich, Schweiz,

2 Finally, we apply the method of Grobner bases to Systems of algebraic équations and to a géométrie problem : Using the "lexicographie ordering" on N", a Grobner basis of an idéal immediately yields idéal bases of the corresponding élimination ideals (see 4.3.). If X is an algebraic subset of the affine rc-space, a Grobner basis with the "inverse lexicographie ordering" permits to obtain an idéal to respect basis of the homogeneous idéal, which dermes the Zariski-closure of X in the projective n-space (see 5.). The method of Grôbner bases was introduced by B. Buchberger in For the history of the theory and for further applications see [B]. Our aim is to give a short and self-contained introduction to the theory of Grobner bases. In this form it could be part of a second or third year algebra course. The results written down in this article can be found elsewhere, but we présent short proofs. We do not enter into questions of implementation or complexity of the algorithms (see for instance [B], [E], [Kl], [T]). Acknowledgements : We thank Bruno Buchberger for sending us a long list of références. We thank Ingrid Mittelberger for her interest and many discussions on this subject. We thank Thierry Vust and the référée for proposing several improvements the flrst version of this article. on 1. Notations and Definitions The notations introduced hère will be valid throughout this article We dénote by R a principal idéal domain (for example: Z, a field, the polynomial ring or power séries ring in one variable over a field) and by #[X] the polynomial ring over R in n variables l9...,x n. Sometimes we make tacitly the additional assumption that we can compute a greatest common divisor of two éléments in R. If S is a subset of #[X], we write <S> for the idéal generated by S X19...,X in [*]. Recall that R[X] is a noetherian ring, this means that every strictly ascending séquence of ideals in R[X] is finite. a 2 n (<X!, For..., ajen" we abbreviate X\" X«... Xl by X a=

3 1.2. Let < be a strict ordering on N" which has the following two properties : a k Well-known examples for such orderings are : the lexicographie ordering (a < (3:^> there is a je{l,...,n} = p k if k <j and a,- < P;), the graded lexicographie ordering L such that the graded inverse lexicographie ordering (a < P:o(X a» GIL i = 1 i = 1 n n < ZPi) or (( pj and there is a;e {1,..., n} such that a k = p fe if k>j and i=l a, > p,))- i=l Examples : As usual, we write a < p instead of (a < p or a = P). Ail expressions like maximum, minimum, smaller,... refer to this ordering Lemma. a) Each agn" is the smallest élément in In particular :ifxa divides X p, then a<p. b) Every strictly descending séquence in N" is finite. In particular, any subset in N" contains a smallest élément. Proof. a) o<y implies a = O+ oc<y + a. b) Let a(l) > a(2) >... be a strictly descending séquence in N". Consider the corresponding séquence Xa(l\Xa(l \ X a{2 \... of monomials. By a) the séquence of ideals <Xa{lXa{l) > a <Xa{l\X a{l \ X«{2) > a... is strictly ascending, hence finite.

4 1.4. With c ax a or c a X a we always tacitly mean that only finitely a aen" many of the coefficients c a are différent from zéro. Let o#p= Ca x a e #[X]. Then we define a If A, B ç= N n, then A + B: = {a + p aea,pe B}. For a subset F ç= jr[x] we define 1.5. Let Jbean idéal in R[X], J# {o}. Définition. A finite subset G of J? {0} is a "Grobner basis of J iff in (G) générâtes the idéal <in (J) >. Remarks and examples. 1) Let R be a field. Then a finite subset G of J? {0} is a Grobner basis of J iff deg (J) (= deg (G) + N"). 2) Grobner bases always exist : Choose a finite generating subset Mç=in (J) of <in(j)>. Then any finite subset G of J with in (G) 3 M is a Grobner basis of J. 3) Not every generating subset of an idéal is a Grobner basis: Consider the graded lexicographie ordering on N 2. Let PP \ x = X\X 2 -\- X and 1 P 2 : = X XX\ be éléments of Q[_X 1,X 2 ~\. Then {i\,p 2} is not a Grobner basis of J:= <Pl,P 1,P 2 >, since X^ = Z 2Pi - X P 1 2 ej, but X 1X 2 i<x\x 2,X 1X 22 2 > = <in(p I ),in(p 2)>. 4) Any finite subset of J? {0} containing a Grobner basis is a Grobner basis. 5) Let Jbe a principal idéal. Then any finite subset of J which contains a generating élément of J is a Grobner basis of J. 6) Any set of monomials {c xx a{l \..., c k Xm } ç R[X] is a Grobner basis of the idéal generated by them Let Jbean idéal in #[X], J {o}. The set in (J) is determined by a "weight-function"

5 where w(ô) is a generating élément of the (principal) idéal So for n = 2 we can visualize in(j) by a figure of the following form: Figure 1. For example, to <2X 1,3X 2>2 > s Z[X 1,X2] corresponds figure 2. Figure 2.

6 If Risa field, then w: deg (J) -+ R is a "weight function". So the corresponding figure is of the form Figure The Division Algorithm Let F be a finite subset of #[X] - {o} Définition. An "admissible combination of F" is an expression of the form L:= c(y, P)X Y P, c(y, P) er, such that Y6N",PeF Example. Let P, ge R[X] and let a, pe N". Then X*P - X p g is an admissible combination of {P, g} iff X a?in (P) # X p? in (g). For every Qg<in (F) > there is an admissible combination L Remark. of F such that in(l) = in (g). L can be calculated in the following way: deg (g) - deg (P) e N"}. Then Let F: = {P e F For PeF' we calculate éléments c(p) er such that le (Q) = SetL:= X c(p)x deg(q) - deg(p) P. PeF' Example: F: = {5X X +1, 3X 2 + 2}, g: = ZjX^ Then L = - J^X^Xi +l) + 2X \X l(3x 2 + 2). c(p) le (P). PeF'