University of Paderborn Department of Mathematics Diploma Thesis ...

3.2. EXPANSIONS AND CONTRACTIONS OF MONOMIALS 47i.e. the monomials of R(x 2 0x 2 ) have to be placed to the right of the monomial x 2 0x 2 in thelexicographic order. Similarly, as one might expect, we get for the elements of L(x 2 0x 2 ):x 3 0 > x 2 0x 1 , x 2 0x 1 > x 2 0x 2 ,i.e. the monomials of L(x 2 0x 2 ) have to be placed to the left of the monomial x 2 0x 2 . Thisexplains the names of the sets.In general, for monomials x A ∈ R and m 1 ∈ R(x A ), m 2 ∈ L(x A ) we have: m 2 > x A > m 1 .Before we come back to the preceding definitions of left-shifts and right-shifts, we have topresent some more technical definitions, which will serve as algorithmic tools later. Nextwe come to define expansions and contractions of monomials.3.2 Expansions and contractions of monomialsIn this section, we will only consider stable and saturated ideals. Thus, if not statedexplicitly, I ⊂ R = K[x 0 , . . . , x n ] always denotes a stable, saturated ideal and I g itsminimal generating set (of monomials).We will now study some techniques, first of all the expansion of a monomial, which dohave an effect on the set of generators of a saturated stable ideal.Definition 3.5. Let I ⊂ R be a saturated stable ideal, and let x A = x a 00 · . . . · x arr ∈ I g ,a r > 0, be one of its minimal generators, such that none of the elements of R(x A ) iscontained in I g (i.e. we have I g ∩ R(x A ) = ∅). Then we call x A expandable in I (or simplyexpandable, if the ideal is clear, we are considering) and the expansion of x A in I is definedto be the ideal I exp , generated by the setI expg := I g \ { x A} ∪ { x A · x r , x A · x r+1 , . . . , x A · x n−1}.Remark 3.6. Let again I ⊂ R be a stable saturated ideal, and let x A = x a 00 · . . . · x arrbe expandable in I.(i) Since I is stable, we get for every monomial x A = x a 00 · . . . · x arr ∈ I g :x a 00 · . . . · x a j−1j−1 · xa j+1j · x a j+1j+1 · . . . · xa k−1k−1 · xa k−1k· x a k+1k+1 · . . . · xar rfor all 0 ≤ j < k ≤ r by Theorem 2.7. Thus, the setL(x A ) = {x a 00 · . . . · x as+1s · x a s+1−1s+1 · . . . · x arr | s = 0, . . . , r − 1}must be contained in I and therefore also in the ideal I exp , since the only element,which is removed from the set I g , is the monomial x A itself and x A is not containedin L(x A ). Thus, to prove the stability of the ideal I exp , it suffices to show that the∈ I g∈ I g