University of Paderborn Department of Mathematics Diploma Thesis ...

18 CHAPTER 1. NOTATIONS AND PREREQUISITESThose vectors, which have been underlined, are still contained in one of the rows above.In the second row, the first vector also appears in the row above, in the third row, the firstvector also appears in the first row and the second vector also appears in the second. In thefourth row, the first vector appears in the first row, the second vector in the second and thethird vector in the third row. Hence, by this combinatorial observation, we conclude thatthe number of monomials of degree d + 1, which can be obtained from a lexicographic setof r monomials of degree d, is the minimal number of monomials, which can be obtainedfrom any set of r monomials of degree d in R. Consequently, any other set of r monomialsof degree d will at least yield as many monomials of degree d + 1 as the set above. Thisindicates that the Hilbert function h L has minimal growth (from degree d up to degree d+1).Hence, the Hilbert function h R/L (j) must have maximal growth. Since the maximal value ofh R/L (d+1) can be at most h R/L (d) 〈d〉 (by Theorem 1.18), we obtain h R/L (d+1) = h R/L (d) 〈d〉 .Example 1.21. Consider the ideals L := (x 2 0, x 0 x 1 , x 0 x 2 ) and I := (x 2 0, x 0 x 1 , x 2 1) inR := K[x 0 , x 1 , x 2 ]. L is lexicographic, because it is generated by the three largest monomialsof degree 2. Since x 0 x 2 > x 2 1 in lexicographic order and x 0 x 2 /∈ I, I is not a lexicographicideal. First, we determine the set of monomials in [L] 3 : Multiplication of x 2 0 by x 0 , x 1 , x 2provides x 3 0, x 2 0x 1 , x 2 0x 2 . The monomial x 0 x 1 yields x 2 0x 1 , x 0 x 2 1, x 0 x 1 x 2 . Finally, we obtainfrom x 0 x 2 the three monomials x 2 0x 2 , x 0 x 1 x 2 and x 0 x 2 2 (those, who appeared before, havebeen underlined as in the remark above). Hence, [L] 3 is generated by the six monomialsx 3 0, x 2 0x 1 , x 2 0x 2 , x 0 x 2 1, x 0 x 1 x 2 , x 0 x 2 2. In contrast to this, [I] 3 is generated by the sevenmonomials x 3 0, x 2 0x 1 , x 2 0x 2 , x 0 x 1 x 2 , x 2 0x 1 , x 3 1, x 2 1x 2 .