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4.2. COMPUTING HILBERT POLYNOMIALS OF STABLE IDEALS 79(iv) Again, we increase the number of variables of R by one, i.e. R := K[x 0 , x 1 , x 2 , x 3 , x 4 ],and consider the ideal I 4 := (x 0 , x 5 1, x 4 1x 3 2, x 4 1x 2 2x 6 3).MuPAD>> compute_Hilbert_series([[1,0,0,0,0],[0,5,0,0,0],[0,4,3,0,0],>> [0,4,2,6,0]], 4);Output15 14 13 12 9 8 7 6 5poly(t - 3 t + 3 t - t - t + 2 t - t + t - t - t + 1, [t]),13 12 7 4 3 2poly(t - t - t + t + t + t + t + 1, [t])Thus, we see that the non-reduced Hilbert series of R/I 4 isand the reduced Hilbert series ist 15 − 3t 14 + 3t 13 − t 12 − t 9 + 2t 8 − t 7 + t 6 − t 5 − t + 1(1 − t) 5t 13 − t 12 − t 7 + t 4 + t 3 + t 2 + t + 1(1 − t) 3 .The next section is dedicated to the computation of Hilbert polynomials of stable ideals.4.2 Computing Hilbert polynomials of stable idealsSince we can compute the Hilbert series of R/I for I ⊂ R an arbitrary stable ideal, we cannow present the source code to compute the Hilbert polynomial of R/I due to Theorem2.27 and Algorithm 2.29: Letk∑g(t) = C i · t i ,g(1) ≠ 0, be the numerator of the reduced Hilbert seriesi=0H R/I (t) =g(t)(1 − t) dof R/I. Then the Hilbert polynomial of R/I can be obtained byk∑( )z + d − 1 − jp R/I (z) = C j ·.d − 1j=0The source code of how to obtain the Hilbert polynomial p R/I (z) is given below. It makesuse of 4.1 to compute the Hilbert series of R/I first.