University of Paderborn Department of Mathematics Diploma Thesis ...

78 CHAPTER 4. ALGORITHMS FOR STABLE IDEALSMuPAD>> compute_Hilbert_series([[2,0,0],[1,1,0],[0,2,0]], 2);Output3 2poly(2 t - 3 t + 1, [t]), poly(2 t + 1, [t])The output has to be understood as follows: The first expression is a univariate polynomialrepresenting the numerator of the non-reduced Hilbert series of R/I 1 in the variablet. The second expression is also a univariate polynomial in the variable t. It represents thenumerator of the reduced Hilbert series. Hence, 2t3 − 3t 2 + 1(1 − t) 3 is the non-reduced Hilbertseries of R/I 1 and 2t + 11 − t is the reduced Hilbert series of R/I 1.(ii) Let I 2 := (x 2 0, x 0 x 1 , x 0 x 2 , x 3 1). The set of generators is encoded byAgain with n = 2 we obtainMuPAD[[2, 0, 0], [1, 1, 0], [1, 0, 1], [0, 3, 0]].>> compute_Hilbert_series([[2,0,0],[1,1,0],[1,0,1],[0,3,0]], 2);Output3 2poly(2 t - 3 t + 1, [t]), poly(2 t + 1, [t])Thus, the non-reduced Hilbert series of R/I 2 is 2t3 − 3t 2 + 1and the reduced Hilbert series(1 − t) 3is given by 2t + 1 as in (i).1 − t(iii) Now, let R := K[x 0 , x 1 , x 2 , x 3 ] and I 3 := (x 0 , x 2 1, x 1 x 2 2, x 4 2). Then, we obtainMuPAD>> compute_Hilbert_series([[1,0,0,0],[0,2,0,0],[0,1,2,0],[0,0,4,0]],3);Output6 5 4 2 3 2poly(- t + t + t - t - t + 1, [t]), poly(t + 2 t + 2 t + 1, [t])i.e. −t6 + t 5 + t 4 − t 2 − t + 1is the non-reduced and t3 + 2t 2 + 2t + 1(1 − t) 4 1 − tseries of R/I 3 .the reduced Hilbert