University of Paderborn Department of Mathematics Diploma Thesis ...

116 CHAPTER 4. ALGORITHMS FOR STABLE IDEALSMuPAD>> compute_all_Hilbert_series(poly(4*z + 1, [z]), 4);Output3 2{poly(3 t + 1, [t]), poly(- t + 2 t + 2 t + 1, [t]),4 3 2poly(- t + t + t + 2 t + 1, [t]),5 4 2poly(- t + t + t + 2 t + 1, [t]),6 4 3poly(- t + t + t + 2 t + 1, [t]),5 4 3 2poly(- t + t + t - t + 3 t + 1, [t]),7 4 3 2poly(- t + t + t + t + t + 1, [t]),6 5 4 3 2poly(- 2 t + t + t + t + t + t + 1, [t])}The above results equal those computed by Alyson Reeves in [16] (see Chapter 3, Example2, pp. 25,26 ).If we want to compute the numerators of the non-reduced Hilbert series as well, we proceedas follows:MuPAD>> compute_all_Hilbert_series(poly(4*z + 1, [z]), 4, All);Output4 3 2{[poly(- 3 t + 8 t - 6 t + 1, [t]), poly(3 t + 1, [t])],6 5 4 3 2[poly(t - 5 t + 7 t - 2 t - t - t + 1, [t]),3 2poly(- t + 2 t + 2 t + 1, [t])], [poly(7 6 5 4 3 2t - 4 t + 5 t - 3 t + 3 t - 2 t - t + 1, [t]),