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A 2-codim polynomial ideal of normal-basis [0], meaning?

asked 2020-08-11 07:22:14 -0600

updated 2020-08-12 02:47:26 -0600

Below is an ideal Id in the ring of polynomials over the rational field. SageMath states that it has codimension 2 but normal-basis [0]. Note that Id.dimension() provides the dimension of the ring modulo the ideal Id.

sage: Id.dimension()
2
sage: B=Id.normal_basis()
sage: B
[0]

Question: How to interpret this result?

Warning: the computations below took 30 min on my laptop

sage: R.<s2,s5,ss5,a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15, a16, a17, a18, a19, a20, a21, a22, a23, a24, a25, a26, a27, a28, b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, c0, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15>=PolynomialRing(QQ)
sage: Id=Ideal(s2^2-2,s5^2-5,ss5^2-s5,a5^2 - 1/4,-a8^2 + 1/5,2*b6^3 - c6^2,2*c6^3 - b6^2,-a4^2 + 1/4,a10^2 - 1/20*s5,-a0^2 + 1/4,-b9*c9 + 1/20*s5,2/5*s5*a17^2 - 1/10,2/5*s5*b1*c1 - 1/5,-a11^2 + 1/20*s5,ss5*b1*c1 - 1/10*ss5^3,a13^2 - 1/20*s5,2/5*s5*a13^2 - 1/10,-b0*c0 + 1/10*s5,a17^2 - 1/20*s5,s2*b3*c3 - 1/10*s5*s2,-a12^2 + 1/20*s5,s2*a4^2 - 1/4*s2,2/5*s5*a10^2 - 1/10,1/2*ss5*s2*b5*c5 - 1/10*ss5*s2,-a6^2 + 1/5,a1^2 - 1/4,s2*b0*c0 - 1/10*s5*s2,-b3*c3 + 1/10*s5,1/2*ss5*s2*b1*c1 - 1/20*ss5^3*s2,1/2*ss5*s2*a6^2 - 1/10*ss5*s2,ss5*b4*c4 - 1/5*ss5,s2*a0^2 - 1/4*s2,2/5*s5*a18^2 - 1/10,-b7*c7 + 1/20*s5,ss5*a6^2 - 1/5*ss5,a3^2 - 1/4,1/2*ss5*s2*b2*c2 - 1/20*ss5^3*s2,-a16^2 + 1/20*s5,a2^2 - 1/4,2/5*s5*a15^2 - 1/10,-b8*c8 + 1/20*s5,a15^2 - 1/20*s5,s2*a1^2 - 1/4*s2,-b4*c4 + 1/5,ss5*b2*c2 - 1/10*ss5^3,a18^2 - 1/20*s5,-a15^2 + 1/20*s5,2/5*s5*a16^2 - 1/10,1/2*ss5*s2*b0*c0 - 1/20*ss5^3*s2,s2*b1*c1 - 1/10*s5*s2,ss5*a8^2 - 1/5*ss5,s2*a5^2 - 1/4*s2,1/2*ss5*s2*b3*c3 - 1/20*ss5^3*s2,-b1*c1 + 1/10*s5,-b5*c5 + 1/5,1/2*ss5*s2*a7^2 - 1/10*ss5*s2,-a18^2 + 1/20*s5,-a7^2 + 1/5,2/5*s5*b2*c2 - 1/5,-a3^2 + 1/4,-a2^2 + 1/4,ss5*b0*c0 - 1/10*ss5^3,ss5*b3*c3 - 1/10*ss5^3,ss5*b5*c5 - 1/5*ss5,a16^2 - 1/20*s5,2/5*s5*a11^2 - 1/10,1/2*ss5*s2*b4*c4 - 1/10*ss5*s2,2/5*s5*b3*c3 - 1/5,1/2*a3^2 - 1/8,-a17^2 + 1/20*s5,-a1^2 + 1/4,s2*a2^2 - 1/4*s2,2/5*s5*b0*c0 - 1/5,1/2*a2^2 - 1/8,a12^2 - 1/20*s5,b6*c6 - 1/4,s2*b2*c2 - 1/10*s5*s2,-a10^2 + 1/20*s5,s2*a3^2 - 1/4*s2,ss5*a7^2 - 1/5*ss5,2/5*s5*a12^2 - 1/10,-a5^2 + 1/4,a11^2 - 1/20*s5,-b10*c10 + 1/20*s5,-b2*c2 + 1/10*s5,-a13^2 + 1/20*s5,a13^2 + a18^2 - 1/10*s5,s5*a17^3 + s5*a18^3,2*a25^3 + 2*a28^3 + 1/25*s5,2*a18^2*a5 + 1/20*s5,1/2*ss5*s2*a26^2 + 1/2*ss5*s2*b15*c15 - 1/10*ss5*s2,2/5*a25 + 2/5*a28 + 1/25*s5,1/2*ss5*s2*a10^2 + 1/2*ss5*s2*b7*c7 - 1/20*ss5^3*s2,a11^2 + b10*c10 - 1/10*s5,1/2*a15 + 1/2*a16,2/5*s5*a13^2 + 2/5*s5*a18^2 - 1/5,a16^2 + b7*c7 - 1/10*s5,-a14*b6,1/5*s5*b1*c1 + 2/5*s5*b8*c8 - 1/5,2/5*s5*a16^2 + 2/5*s5*b7*c7 - 1/5,2*a0*a3^2 - a3^2,1/2*ss5*s2*b12*c12 + 1/2*ss5*s2*b13*c13 - 1/10*ss5*s2,2*a3^3 - a0*a3,1/2*ss5*s2*a17^2 + 1/2*ss5*s2*a18^2 - 1/20*ss5^3*s2,2*a22^3 + 2*a23^3 + 1/25*s5,2/5*s5*a22^2 + 2/5*s5*a23^2 - 3/50*s5,a10^2 + a15^2 - 1/10*s5,1/2*ss5*s2*a25^2 + 1/2*ss5*s2*b14*c14 - 1/10*ss5*s2,b10*c10 + 1/2*b3*c3 - 1/10*s5,a4^3 + 2*a5^3 + 1/8,-a9*b6,2*a13^2*a9 - a13^2 + 1/20*s5,1/2*ss5*s2*b10*c10 + 1/2*ss5*s2*b9*c9 - 1/20*ss5^3*s2,a2^2*a5 + 1/8,1/2*ss5*s2*a26*b15 + 1/2*ss5*s2*a26*c15,2/5*s5*a17^2 + 2/5*s5*b8*c8 - 1/5,1/2*b2*c2 + b9*c9 - 1/10*s5,a2^2 + a3^2 - 1/2,1/2*a2 + 1/2*a3,1/2*ss5*s2*a24^2 + 1/2*ss5*s2*b14*c14 - 1/10*ss5*s2,2/5*s5*a19^2 + 2/5*s5*a20^2 - 3/50*s5,2/5*s5*b14*c14 + 2/5*s5*b15*c15 - 1/10*s5,2/5*s5*a10^2 + 2/5*s5*a15^2 - 1/5,2/5*s5*b10*c10 + 2/5*s5*b8*c8 - 1/5,a0^3 + 2*a1^3 + 1/8,2*a10^2*a9 - a10^2 + 1/20*s5,2*a14*a18^2 - a18^2 + 1/20*s5,1/2*ss5*s2*a20^2 + 1/2*ss5*s2*b14*c14 - 1/10*ss5*s2,b7*c7 + b9*c9 - 1/10*s5,2/5*s5*b7*c7 + 2/5*s5*b9*c9 - 1/5,a2^2*a4 - a2^2 + 1/8,1/2*a11 + 1/2*a13,1/2*a17 + 1/2*a18,2/5*a22 + 2/5*a23 + 1/25*s5,2*a1*a11^2 + 1/20*s5,b10*c10 + b8*c8 - 1/10*s5,2*a1*a13^2 + 1/20*s5,s5*a15^3 + s5*a16^3,1/2*ss5*s2*a12^2 + 1/2*ss5*s2*b8*c8 - 1/20*ss5^3*s2,1/2*b0*c0 + b7*c7 - 1/10*s5,1/2*ss5*s2*a16^2 + 1/2*ss5*s2*b10*c10 - 1/20*ss5^3*s2,1/2*ss5*s2*b2*c2 + 1/2*ss5*s2*b3*c3 - 1/10*ss5^3*s2,2/5*s5*a17*a18 + 1/10,2*a14*a15^2 - a15^2 + 1/20*s5,2/5*s5*b11*c11 + 2/5*s5*b14*c14 - 1/10*s5,1/2*ss5*s2*a27^2 + 1/2*ss5*s2*b15*c15 - 1/10*ss5*s2,2/5*s5*a11*a13 + 1/10,2/5*s5*a25^2 + 2/5*s5*a28^2 - 3/50*s5,1/2*ss5*s2*a18^2 + 1/2*ss5*s2*b10*c10 - 1/20*ss5^3*s2,1/2*ss5*s2*a24^2 + 1/2*ss5*s2*b12*c12 - 1/10*ss5*s2,2*a14^3 + a5^3 + 1/8,1/2*ss5*s2*b7*c7 + 1/2*ss5*s2*b8*c8 - 1/20*ss5^3*s2,2/5*s5*b10*c10 + 1/5*s5*b3*c3 - 1/5,1/2*ss5*s2*b0*c0 + 1/2*ss5*s2*b1*c1 - 1/10*ss5^3*s2,2/5*s5*a24^2 + 2/5*s5*a26^2 - 3/50*s5,2/5*s5*a12^2 + 2/5*s5*b9*c9 - 1/5,2/5*s5*a10*a12 + 1/10,a1^3 + 2*a9^3 + 1/8,1/2*ss5*s2*b13*c13 + 1/2*ss5*s2*b14*c14 - 1/10*ss5*s2,a1*a3^2 + 1/8,1/2*ss5*s2*a19^2 + 1/2*ss5*s2*b11*c11 - 1/10*ss5*s2,a11*a13 + 1/20*s5,a0*a3^2 - a3^2 + 1/8,1/2*ss5*s2*a23^2 + 1/2*ss5*s2*b12*c12 - 1/10*ss5*s2,2*a16^2*a5 + 1/20*s5,a9^2 + b6*c6 - 1/4,1/2*ss5*s2*a13^2 + 1/2*ss5*s2*b8*c8 - 1/20*ss5^3*s2,2*a22^3 + 2*a25^3 + 1/25*s5,1/2*ss5*s2*a15^2 + 1/2*ss5*s2*a16^2 - 1/20*ss5^3*s2,a17*a18 + 1/20*s5,2*a9*b6*c6,2*a1*a10^2 + 1/20*s5,a10*a12 + 1/20*s5,1/4*a4 + 1/2*a5 + 1/8,a14^2 + b6*c6 - 1/4,1/2*a4^2 + a5^2 - 3/8,1/2*ss5*s2*a19*b11 + 1/2*ss5*s2*a19*c11,2*a17^2*a5 + 1/20*s5,2*a2^3 + 2*a3^3,2/5*s5*a22^2 + 2/5*s5*a25^2 - 3/50*s5,2*a14*b6*c6,1/2*ss5*s2*a28^2 + 1/2*ss5*s2*b15*c15 - 1/10*ss5*s2,2*a1*a12^2 + 1/20*s5,2/5*s5*a15*a16 + 1/10,1/4*a0 + 1/2*a1 + 1/8,1/2*ss5*s2*a17^2 + 1/2*ss5*s2*b9*c9 - 1/20*ss5^3*s2,1/5*s5*b0*c0 + 2/5*s5*b7*c7 - 1/5,1/2*b1*c1 + b8*c8 - 1/10*s5,2*a15^2*a5 + 1/20*s5,1/2*a0^2 + a1^2 - 3/8,2*a23^3 + 2*a28^3 + 1/25*s5,2/5*a23 + 2/5*a28 + 1/25*s5,2/5*s5*a6^2 + 2/5*s5*a7^2 - 4/25*s5,1/2*ss5*s2*a11^2 + 1/2*ss5*s2*b7*c7 - 1/20*ss5^3*s2,a17^2 + b8*c8 - 1/10*s5,1/2*ss5*s2*a11^2 + 1/2*ss5*s2*a13^2 - 1/20*ss5^3*s2,2/5*a22 + 2/5*a25 + 1/25*s5,a12^2 + b9*c9 - 1/10*s5,1/2*ss5*s2*a20^2 + 1/2*ss5*s2*b12*c12 - 1/10*ss5*s2,2*a9*b6^2,1/2*a2*a3 + 1/8,1/2*a10 + 1/2*a12,2/5*s5*b12*c12 + 2/5*s5*b15*c15 - 1/10*s5,1/2*ss5*s2*a15^2 + 1/2*ss5*s2*b9*c9 - 1/20*ss5^3*s2,1/5*s5*b2*c2 + 2/5*s5*b9*c9 - 1/5,1/2*ss5*s2*a21^2 + 1/2*ss5*s2*b11*c11 - 1/10*ss5*s2,2*a2^2*a4 - a2^2,a15*a16 + 1/20*s5,2*a2^3 - a2*a4,1/2*ss5*s2*a10^2 + 1/2*ss5*s2*a12^2 - 1/20*ss5^3*s2,s5*a10^3 + s5*a12^3,2/5*s5*a11^2 + 2/5*s5*b10*c10 - 1/5,1/2*a14 + 1/4*a5 + 1/8,a14^2 + 1/2*a5^2 - 1/8,s5*a11^3 + s5*a13^3,2/5*s5*a23^2 + 2/5*s5*a28^2 - 3/50*s5,2*a14*b6^2,1/2*ss5*s2*a22^2 + 1/2*ss5*s2*b11*c11 - 1/10*ss5*s2,1/4*a1 + 1/2*a9 + 1/8,2/5*s5*b11*c11 + 2/5*s5*b12*c12 - 1/10*s5,1/2*a1^2 + a9^2 - 1/8,2*b11*c0^2 - b4*c11,a1*a2^2 - a9^2 + 1/8,s5*c15*c3*c5 - b15*b3,2*a11^2*a23 - a19^2 + 1/10,2*a0*a1*a2 - a1*a2,2*a19^3 + 2*a24^3 + a6^3,2*a14*a16^2 - a17^2 + 1/20*s5,2*a13*a18*b6 - a13*a18,s5*b14*b2*b4 - c14*c2,2*c1^2*c11 - b11*b4,2*a2^2*a9 - a1*a9,2*a1*a2^2 - a0*a1,s5*b2^2*c3 - a4*b3,2*a3^2*a5 - a4*a5,1/2*ss5*s2*a23*c12 + 1/2*ss5*s2*b12*c13,1/2*ss5*s2*a27*b15 + 1/2*ss5*s2*a28*c15,s5*b1^2*c0 - a0*b0,1/2*ss5*s2*b14*c13 + 1/2*ss5*s2*a25*c14,2*b14*c1^2 - b5*c14,2*c14*c2^2 - b14*b4,2*a13*a18*c6 - a13*a18,2*a14*a3*a5 - a14*a3,2*a17^2*a22 - a24^2 + 1/10,a0*a2*a3 + 1/8,1/2*a0*a1 + a1*a9 + 1/8,a0^2*a1 + 2*a1^2*a9 + 1/8,2*a10^2*a28 - a23^2 + 1/10,2/5*a20 + 2/5*a26 + 1/5*a7,2*a12*a13*a7 + 1/10,2*a12^2*a25 - a19^2 + 1/10,1/5*s5*a8^2 + 2/5*s5*b4*c4 + 2/5*s5*b5*c5 - 1/5*s5,1/2*ss5*s2*a28*b15 + 1/2*ss5*s2*a27*c15,2*b10*b6*c7 - b7*c10,2*b7*c10*c6 - b10*c7,2/5*s5*a21^2 + 2/5*s5*b13*c13 + 1/5*s5*b4*c4 - 1/10*s5,s5*b15*b3*b5 - c15*c3,s5*b0*b4*c11 - b11*c0,2/5*a19 + 2/5*a24 + 1/5*a6,s5*b1*b11*b4 - c1*c11,2*a16^2*a22 - a24^2 + 1/10,a0*a2^2 - a1^2 + 1/8,2*a18^2*a22 - a23^2 + 1/10,2*a14*a3^2 - a14*a5,s5*b12*b3*b4 - c12*c3,2*a3*a5^2 - a3*a4,2*a4*b2^2 - b3^2,2*a11^2*a28 - a20^2 + 1/10,2*a0*b1*c0 - b0*c1,s5*b1*b5*c14 - b14*c1,a4^2*a5 + 2*a14*a5^2 + 1/8,a14*a5 + 1/2*a4*a5 + 1/8,2*b12*c0^2 - b5*c12,2*a12^2*a28 - a20^2 + 1/10,2/5*s5*a19^2 + 2/5*s5*a24^2 + 1/5*s5*a6^2 - 1/10*s5,s5*b2*b5*c15 - b15*c2,2*a4*b3*c2 - b2*c3,2*a15*a17*a6 + 1/10,2*a3*a4*a5 - a3*a5,2*c15*c3^2 - b15*b5,s5*b14*c1*c5 - b1*c14,2*a1*a2*a9 - a2*a9,2*a13^2*a28 - a25^2 + 1/10,2*a10*a15*b6 - a10*a15,s5*b0*b5*c12 - b12*c0,2*a15^2*a22 - a25^2 + 1/10,2*a10*a15*c6 - a10*a15,a3^2*a4 - a5^2 + 1/8,2*a16*a18*a6 + 1/10,2*a18^2*a25 - a28^2 + 1/10,2*a12^2*a9 - a11^2 + 1/20*s5,2*a13^2*a23 - a22^2 + 1/10,2*a11^2*a9 - a12^2 + 1/20*s5,a14*b6 + a9*c6,2*c12*c3^2 - b12*b4,s5*c1*c11*c4 - b1*b11,1/2*ss5*s2*a25*b14 + 1/2*ss5*s2*b13*c14,2/5*s5*a27^2 + 2/5*s5*b13*c13 + 1/5*s5*b5*c5 - 1/10*s5,2*a10^2*a25 - a22^2 + 1/10,2*a16^2*a23 - a26^2 + 1/10,a3^2*a5 - a14^2 + 1/8,s5*b0^2*c1 - a0*b1,2/5*s5*a20^2 + 2/5*s5*a26^2 + 1/5*s5*a7^2 - 1/10*s5,1/2*ss5*s2*a21*b11 + 1/2*ss5*s2*a22*c11,s5*b3^2*c2 - a4*b2,2*a14*a17^2 - a16^2 + 1/20*s5,s5*c12*c3*c4 - b12*b3,2*a10*a11*a7 + 1/10,1/2*ss5*s2*a22*b11 + 1/2*ss5*s2*a21*c11,2*a4*c3^2 - c2^2,2*a0*c1^2 - c0^2,2*b15*c2^2 - b5*c15,a2*a3*a4 + 1/8,2*a15^2*a23 - a28^2 + 1/10,2*a20^3 + 2*a26^3 + a7^3,2*a17^2*a25 - a26^2 + 1/10,2*a0*b0^2 - b1^2,1/2*ss5*s2*a23*b12 + 1/2*ss5*s2*b13*c12,2*a1^2*a2 - a0*a2,2*b6*b8*c9 - b9*c8,s5*a10*a16^2 + s5*a12*b10*c10,2*a11*a13*a9 - b8*c8 + 1/20*s5,1/2*b3*c2 + b10*c9,2*a12*a13*a26 - b14*c14 + 1/10,s5*a19*a6*b1 - a22*b1,2*a23*a24^2 + 2*a26^2*a28 + 1/25*s5,s5*a11*b7*c7 + s5*a13*b8*c8,2*a10*a13*a26 - a24^2 + 1/10,2*a14*a18^2 + 2*a13^2*c6 - a18^2,2*a17*a18*a22 - b12*c12 + 1/10,s5*a22*a6*c0 - a19*c0,2*a17*a18*a25 - b15*c15 + 1/10,2*a10^2*a2 + 2*a15^2*a3,2*a22*b11*c11 + 2*a23*b12*c12,2*a20*b12*c12 + 2*a26*b15*c15,2*a28*b2*c2 - a26*a7,2*a5*b6*c6 - a2*a3,2*a3*b6*c6 - a2*a5,2/5*s5*a10*c7 + 2/5*s5*a15*c9,2*a19*b11*c11 + 2*a24*b14*c14,2*a16^2*a3 + 2*a2*b7*c7,2*a21*b11^2 + 2*b12^2*c13,2*a18*a5*c3 - a17*c3,2*a10*a12*a28 - b12*c12 + 1/10,s5*a13*a6^2 - a10*a7,s5*a10*a6^2 - a13*a7,1/2*ss5*s2*a17*c8 + 1/2*ss5*s2*a12*c9,s5*a12^2*a15 + s5*a16*b8*c8,2*a1*a11*b1 - a13*b1,s5*a18*b3*c3 - a17*a5,2*a10^2*a23 + 2*a15^2*a28 - a23*a28,2*a10*a11*a26 - b12*c12 + 1/10,a8*b2^2 + 2*b9^2*c4,2*a10*a12*a9 - b7*c7 + 1/20*s5,s5*a7*c15*c2 - b15*c3,2*a23*b0*c0 - a20*a7,2*a20*b0*c0 - a23*a7,2*a15*a16*a22 - b14*c14 + 1/10,1/2*b1*c0 + b7*c8,1/5*s5*b3*c2 + 2/5*s5*b10*c9,s5*a11*b1*c1 - a1*a13,2*a19^2*a22 + 2*a20^2*a25 + 1/25*s5,s5*a15*b2*c2 - a16*a5,s5*a20*a7*c0 - a23*c0,s5*a26*a7*c2 - a28*c2,2*a2*a5*b6 - a3*b6,s5*a24*a6*b3 - a23*b3,1/2*ss5*s2*b0*b7 + 1/2*ss5*s2*b1*b8,2*a10*a11*a20 - b11*c11 + 1/10,2*a10*a13*a7 - a6^2 + 1/10,2*a14*a15*a16 - b9*c9 + 1/20*s5,2*a15*a18*a6 - a7^2 + 1/10,2*a10^2*a22 + 2*a15^2*a25 - a22*a25,a8*b0^2 + 2*c5*c7^2,2*a15*a18*a24 - a26^2 + 1/10,a10*c7 + a15*c9,1/2*ss5*s2*a17*b8 + 1/2*ss5*s2*a12*b9,2/5*s5*a22*a23 + 2/5*s5*a25*a28 + 1/25*s5,s5*a12*b0*c0 - a1*a10,2*a2*a3*b6 - a5*b6,1/2*ss5*s2*a11*b10 + 1/2*ss5*s2*a16*b7,2*a19^2*a22 + 2*a20^2*a23 + 1/25*s5,s5*a23*a7*c0 - a20*c0,2*a24^2*a25 + 2*a26^2*a28 + 1/25*s5,2*a13^2*a9 + 2*a18^2*c6 - a13^2,s5*a15*a7^2 - a18*a6,2*a21*c11^2 + 2*b14^2*c13,2*a11^2*a2 + 2*a3*b10*c10,s5*a10*a15^2 + s5*a12*b9*c9,2*a17*a5*b3 - a18*b3,2*a26*b2*c2 - a28*a7,2*a14^2*a5 + a4*a5^2 - a3^2 + 1/8,2*a25*b14*c14 + 2*a28*b15*c15,s5*a7*b15*b2 - b3*c15,2*a1*a10*c0 - a12*c0,2*a1*a12*c0 - a10*c0,s5*a16*b2*c2 - a15*a5,2*a26*b3*c3 - a28*a7,2*a13^2*a2 + 2*a18^2*a3,2*a23*a6^2 + 2*a28*a7^2 + 1/25*s5,a0*b1*c1 + 2*a1*b8*c8,2*b13*c12^2 + 2*a27*c15^2,a0*b0*c0 + 2*a1*b7*c7,2*a22*b0*c0 - a19*a6,2*a22*a6^2 + 2*a23*a7^2 + 1/25*s5,s5*a18*b10*c10 + s5*a17*b9*c9,s5*a7*b15*c3 - c15*c2,2*a1*a3*b6 - a2*b6,s5*a17*a6*c5 - a16*c5,2/5*s5*b14*b15 + 2/5*s5*b12*c11,2*b15*b2*c3 - a7*c15,2*a15*a16*a23 - b15*c15 + 1/10,2*a16*a18*a19 - b12*c12 + 1/10,s5*a17*b3*c3 - a18*a5,2*a11*a12*a20 - a21^2 + 1/10,s5*a26*a7*b3 - a28*b3,s5*a13^3 + s5*a11*b8*c8 - a13*a9,2*a14*a17*a18 - b10*c10 + 1/20*s5,s5*a25*a7*c1 - a20*c1,s5*a28*a7*c2 - a26*c2,2*a2*b7*c7 + 2*a3*b9*c9,a8*b1^2 + 2*c5*c8^2,2*a25*b1*c1 - a20*a7,a8*b3^2 + 2*b10^2*c4,2*a28*b3*c3 - a26*a7,s5*a25*a6*c2 - a24*c2,2*a16*a17*a24 - a27^2 + 1/10,s5*a13^2*a18 + s5*a17*b8*c8,a4*b2*c2 + 2*a5*b9*c9,s5*a6*c0*c11 - b11*c1,1/2*ss5*s2*a10*a15 + 1/2*ss5*s2*b9*c7,2*a17*a18*a5 - b3*c3 + 1/20*s5,2*a5*b10*c10 + a4*b3*c3,s5*a18^3 + s5*a17*b10*c10 - a14*a18,s5*a10^3 + s5*a12*b7*c7 - a10*a9,2*a11*a12*a7 - b4*c4 + 1/10,1/2*ss5*s2*b3*c10 + 1/2*ss5*s2*b2*c9,s5*a10*b0*c0 + s5*a12*b1*c1,2*a3*b10*c10 + 2*a2*b8*c8,2*a12^2*a2 + 2*a3*b9*c9,2*b1*c0*c11 - a6*b11,1/5*s5*b1*c0 + 2/5*s5*b7*c8,s5*a17*b2*c2 + s5*a18*b3*c3,2*a1*a11*a13 - b1*c1 + 1/20*s5,1/2*ss5*s2*a20*b12 + 1/2*ss5*s2*a24*b14,s5*a20*a7*b1 - a25*b1,1/2*ss5*s2*a13*a18 + 1/2*ss5*s2*b10*c8,2*a10^2*a9 + 2*a15^2*c6 - a10^2,2*a15*a17*a19 - b14*c14 + 1/10,2/5*s5*b11*c12 + 2/5*s5*c14*c15,2/5*s5*a18*c10 + 2/5*s5*a13*c8,2*a11*a13*a23 - b11*c11 + 1/10,1/2*ss5*s2*c0*c7 + 1/2*ss5*s2*c1*c8,1/2*ss5*s2*a24*c12 + 1/2*ss5*s2*a20*c14,1/2*ss5*s2*a11*c10 + 1/2*ss5*s2*a16*c7,s5*a10*b7*c7 + s5*a12*b8*c8,s5*a23*a6*c3 - a24*c3,2/5*s5*a19*a24 + 2/5*s5*a20*a26 + 1/25*s5,2/5*s5*a24*a6 + 2/5*s5*a26*a7 + 1/25*s5,2/5*s5*a19*a6 + 2/5*s5*a20*a7 + 1/25*s5,a2*a3*a5 - b6*c6 + 1/8,a1*a2*a3 - b6*c6 + 1/8,1/2*ss5*s2*a10*a15 + 1/2*ss5*s2*b7*c9,s5*a10*a6*a7 - a13*a6,2*a10*a12*a25 - b11*c11 + 1/10,a12*c8 + a17*c9,s5*a10^2*a15 + s5*a16*b7*c7,s5*a16*b5*c5 - a17*a6,2*b0*b11*c1 - a6*c11,2*a19*b0*c0 - a22*a6,s5*a13*b1*c1 - a1*a11,2*a17^2*a3 + 2*a2*b8*c8,2*a16*a18*a24 - b15*c15 + 1/10,2*a1*b6*c6 - a2*a3,2*a23*b12*c12 + 2*a28*b15*c15,s5*a17*b5*c5 - a16*a6,2*a11*a13*a28 - b14*c14 + 1/10,a18*c10 + a13*c8,2*a1*a2*c6 - a3*c6,s5*a12*a7*c4 - a11*c4,s5*a15*a6*a7 - a18*a7,2/5*s5*a12*b8 + 2/5*s5*a17*b9,s5*a19*a6*c0 - a22*c0,1/2*ss5*s2*b9*c2 + 1/2*ss5*s2*b10*c3,2*a20*b1*c1 - a25*a7,s5*a22*a6*c1 - a19*c1,2*a16*a17*a19 - b13*c13 + 1/10,2*a15*a17*a24 - b15*c15 + 1/10,2*a13^2*a22 + 2*a18^2*a23 - a22*a23,2*a25*a6^2 + 2*a28*a7^2 + 1/25*s5,1/2*ss5*s2*a13*a18 + 1/2*ss5*s2*b8*c10,a0*a1^2 + 2*a1*a9^2 - a2^2 + 1/8,2*b3*c15*c2 - a7*b15,2*a20*b14*c14 + 2*a26*b15*c15,2*a15*a16*a5 - b2*c2 + 1/20*s5,2/5*s5*b15*c12 + 2/5*s5*c11*c14,2*a16*a5*c2 - a15*c2,s5*a15*b2*c2 + s5*a16*b3*c3,2*a2*a3*b6 - a1*b6,s5*a6*b0*b11 - b1*c11,2*a2*b6*c6 - a1*a3,s5*a18*a7^2 - a15*a6,2/5*s5*b11*b14 + 2/5*s5*b12*c15,s5*a12*b4*c4 - a11*a7,s5*a24*a6*c2 - a25*c2,s5*a13*a17^2 + s5*a11*b9*c9,2*a15*a18*a19 - a20^2 + 1/10,s5*a16*a6*c5 - a17*c5,2*a1*a10*a12 - b0*c0 + 1/20*s5,s5*a11*a7*c4 - a12*c4,2*a14*a15^2 + 2*a10^2*c6 - a15^2,2/5*s5*a22*a25 + 2/5*s5*a23*a28 + 1/25*s5,2*a3*a5*c6 - a2*c6,2*a19*b11*c11 + 2*a24*b12*c12,s5*a10*b0*c0 - a1*a12,s5*a13*a6*a7 - a10*a6,2*a19*b1*c1 - a22*a6,s5*a15^3 + s5*a16*b9*c9 - a14*a15,2*a27*b15^2 + 2*b13*c14^2,s5*a11*b0*c0 + s5*a13*b1*c1,s5*a16*b10*c10 + s5*a15*b9*c9,s5*a18*a6*a7 - a15*a7,2*a12*a13*a20 - b11*c11 + 1/10,2*a22*b1*c1 - a19*a6,2*a10*a13*a20 - a19^2 + 1/10,a16*c10 + a11*c7,s5*a6*b11*c1 - c0*c11,2*a1*a13*c1 - a11*c1,1/2*ss5*s2*a20*c12 + 1/2*ss5*s2*a24*c14,2*a13^2*a25 + 2*a18^2*a28 - a25*a28,2*a25*b2*c2 - a24*a6,2*a24*b2*c2 - a25*a6,2*a23*b3*c3 - a24*a6,2*a22*b11*c11 + 2*a25*b14*c14,s5*a11*b4*c4 - a12*a7,2*a22*a6^2 + 2*a25*a7^2 + 1/25*s5,2*a24*b3*c3 - a23*a6,2/5*s5*a16*c10 + 2/5*s5*a11*c7,s5*a13*a18^2 + s5*a11*b10*c10,1/2*ss5*s2*a24*b12 + 1/2*ss5*s2*a20*b14,s5*a28*a7*c3 - a26*c3,2*a16*a17*a6 - b5*c5 + 1/10,s5*a11^2*a18 + s5*a17*b7*c7,2*a15*a5*b2 - a16*b2,2*a11*a12*a26 - b13*c13 + 1/10,2*b13*c0*c1 - a24*b5,2*a27*c12^2 + 2*c11^2*c13 - b7^2,2*a11*b10*c6 - a12*b9,2*a14*c10^2 + 2*c6*c8^2 - c9^2,s5*a11*b7*c10 + s5*a10*b7*c9,s5*b2*c1*c8 - a12*a2,2*a11*a2*c0 - b7*c3,2*b3*c12*c2 - a6*c14,s5*a8*c0*c4 - a6*b1,2*a2*b0*b7 - a11*b3,2*a17*b10*c11 - b12*c13,2*a16^2*a24 + 2*a19*b7*c7 - a22*a24,2*a14^2*a3 + 2*a2*b6*c6 - a3*a5,2*a22^2*a23 + 2*a25^2*a28 - a13^2 + 1/25*s5,a3*b3*c0 - a16*b10,2*a5*c10*c3 - c2*c9,2*a11*b10*b6 - a18*b8,s5*a17*a18*b10 + s5*b9^2*c10 - a14*b10,s5*a26*b2*c15 + s5*a27*b3*c15,s5*a26*b2*b5 - a27*c3,2*a11^2*a19 + 2*a24*b10*c10 - a19*a23,2*b10^2*b13 + b3^2*b5 - c15^2,s5*b4*c5*c9 - a7*b10,s5*a7*c5*c9 - b10*c4,s5*a10*b0*c2 - a2*c7,s5*a11*b3*c0 - a2*b7,2*a13*c15*c8 - a25*b14,s5*a6*c0*c4 - a8*b1,2*a10*c14*c7 - a22*b11,s5*a23*b0*c12 + s5*a24*b1*c12,2*a7*c2*c3 - a8*b5,s5*a26*b15*b2 + s5*a28*b15*b3,2*a13*a2*c1 - b8*c3,s5*a6*c1*c4 - a8*b0,2*a12*c14*c8 - a19*c11,2*a12*a2*b1 - b2*c8,s5*a11*a12^2 + s5*a13*b8*c8 - a11*a9,s5*a16^2*a17 + s5*a18*b10*c10 - a14*a17,2*a16*b10*b12 - a26*c15,2*a1*b0*c7 - b1*c8,s5*a6*b12*c3 - b14*c2,2*a11*a18*c6 - b10*c8,a3*b2*c0 - a15*b9,2*a15*b12*b9 - a28*b15,2*a1*c1*c7 - c0*c8,2*a6*c11*c12 + 2*a7*c12*c15,s5*a27*b15*b2 + s5*a26*b15*b3,2*a13*a17*c6 - b8*c10,2*a5*b3*c9 - b2*c10,2*a3*c0*c9 - a15*c2,2*a13*c12*c8 - a22*c11,s5*b1*b8*c3 - a13*a2,2*b6*b7*c9 - a12*a15,s5*a16*b7*c10 + s5*a18*b8*c10,s5*b1*b7*c0 - a1*b8,s5*a7*c3*c5 - a8*b2,2*a13*b15*b7 - a24*c14,s5*b1*b9*c2 - a17*a3,s5*a11^2*a12 + s5*a10*b7*c7 - a12*a9,s5*b0*b8*c1 - a1*b7,s5*a18*a28^2 + s5*a17*b15*c15 - a18*a25,2*a18*c14*c9 - a26*b15,2*a10*a16*b6 - b7*c9,s5*a7*c1*c14 - c0*c12,2*a11*c6*c9 - a12*c10,s5*a16*a26^2 + s5*a15*b15*c15 - a16*a23,2*a17*b6*b8 - a13*b10,a2*b1*c2 - a12*c8,s5*a11*a20^2 + s5*a13*b14*c14 - a11*a28,2*a15*b11*b9 - a25*c14,2*a22*a25^2 + 2*a23*a28^2 - a15^2 + 1/25*s5,2*a12*b14*b7 - a21*b11,2*c0*c1*c4 - a6*a8,2*a3*c1*c9 - a17*c2,s5*a6*c5*c7 - b8*c4,2*a16*b6*b7 - a17*b8,2*a17^2*a24 + 2*a19*b8*c8 - a22*a24,2*a17*b6*b7 - a16*b8,s5*a10*a23^2 + s5*a12*b12*c12 - a10*a28,s5*a21*b1*b4 - a19*c0,2*a13*a2*c3 - c1*c8,s5*a15*a25^2 + s5*a16*b14*c14 - a15*a22,s5*b3*c0*c10 - a16*a3,s5*a16*a24^2 + s5*a15*b14*c14 - a16*a22,2*a22*a24^2 + 2*a25*a26^2 - a17^2 + 1/25*s5,2*a15*a3*c0 - b9*c2,2*a2*b3*c8 - a13*b1,2*a15*b10*b11 - a20*c14,s5*a19*b1*b4 - a21*c0,2*a22*a24^2 + 2*a23*a26^2 - a16^2 + 1/25*s5,2*a12*b7*c15 - b12*c13,2*a17*a3*c1 - b9*c2,s5*a15*a28^2 + s5*a16*b15*c15 - a15*a23,s5*a13*b8*c10 + s5*a12*b8*c9,2*a26*c2*c3 - a27*b5,2*a17*b10*b14 - a27*b15,2*a10*b15*c7 - a23*b12,2*a2*b7*c2 - a10*c0,s5*b10*b5*c4 - a7*c9,2*a3*b1*b10 - a18*b3,b0^2*b5 + 2*a27*c7^2 - b12^2,s5*a15*a16*b9 + s5*b10^2*c9 - a14*b9,2*a21*c0*c1 - a19*b4,s5*b10*b3*c2 - a5*b9,2*a19*c0*c1 - a21*b4,2*a13*b12*b7 - a19*b11,2*a19^2*a25 + 2*a20^2*a28 - a12^2 + 1/25*s5,2*a11*c12*c7 - a19*b11,2*a2*c0*c7 - a10*c2,2*a10*a2*b0 - b2*c7,2*a12*c6*c9 - a11*c10,2*a3*b10*c3 - a16*c0,s5*a7*a8*b3 - c2*c5,2*a11*c15*c7 - a20*b12,2*b8*c10*c6 - a11*a18,2*a18*b11*c9 - a20*b12,s5*b5*b8*c4 - a6*c7,s5*a11*a13*b8 + s5*b7^2*c8 - a9*b8,2*a19*c11^2 + 2*a20*b14*c12,2*a13*b6*c10 - a17*c8,2*b6*b9*c7 - a10*a16,b0^2*b4 + 2*c13*c7^2 - b11^2,s5*a6*a8*b1 - c0*c4,2*a10*c14*c8 - a19*b11,s5*a21*c0*c4 - a19*b1,s5*a7*c2*c5 - a8*b3,s5*c13*c2*c4 - a20*b3,2*a3*b3*c10 - a18*b1,2*b11^2*c13 + 2*a27*c14^2 - b8^2,2*a12*a2*b2 - b1*b8,s5*a15*b7*c9 + s5*a17*b8*c9,s5*a6*c14*c3 - c12*c2,2*a2*b1*b8 - a13*b3,2*a14*a3*b6 + 2*a2*a9*c6,2*a11*c12*c8 - a21*b11,s5*a7*b14*c0 - b12*c1,s5*a19*b0*b4 - a21*c1,2*a15*a3*c2 - c0*c9,s5*b2*c0*c7 - a10*a2,s5*a19*b0*c11 + s5*a21*b1*c11,s5*a13*a25^2 + s5*a11*b14*c14 - a13*a28,s5*a16*b0*c3 - a3*c10,2*b11^2*c4 + 2*b12^2*c5 - b0^2,2*a17*a3*c2 - c1*c9,s5*a18*a23^2 + s5*a17*b12*c12 - a18*a22,s5*a17*a26^2 + s5*a18*b15*c15 - a17*a25,a2*b2*c0 - a10*b7,s5*a7*b10*c4 - c5*c9,s5*a8*b3*b5 - a7*c2,s5*a7*a8*b2 - c3*c5,s5*a12*a20^2 + s5*a10*b12*c12 - a12*a28,s5*a27*c2*c5 - a26*b3,2*a12^2*a19 + 2*a24*b9*c9 - a19*a25,s5*b4*c5*c7 - a6*b8,2*a21*b10^2 + b3^2*b4 - c12^2,2*a24*c0*c1 - b5*c13,a3*b3*c1 - a18*b10,2*b6*b8*c10 - a13*a17,2*a16*a3*b0 - b3*c10,2*a14*b9^2 + 2*b7^2*c6 - b10^2,s5*a25*b14*b2 + s5*a20*b14*b3,2*a18*b6*c8 - a11*c10,b1^2*b4 + 2*c13*c8^2 - c11^2,2*b2*c14*c3 - a6*c12,2*a18*a3*c1 - b10*c3,2*a10*c6*c9 - a16*c7,s5*b0*b7*c3 - a11*a2,s5*a7*c10*c5 - b9*c4,s5*a26*b3*b5 - a27*c2,2*a2*a9^2 + 2*a3*b6*c6 - a1*a2,2*a12*b6*c9 - a15*c7,s5*b1*b5*c13 - a24*c0,2*a3*c0*c10 - a16*c3,2/5*s5*a21*b4 + 1/5*s5*a8*b4 + 2/5*s5*b5*c13,2/5*s5*a21*b13 + 2/5*s5*a27*b13 + 1/5*s5*b5*c4,s5*b13*b3*b4 - a20*c2,2*a18*a3*c3 - c1*c10,2*a17*b6*c8 - a16*c7,2*a16*b6*c8 - a17*c7,2*a11*c15*c8 - b13*c14,2*b0*b12*c1 - a7*b14,s5*a15*b0*c2 - a3*c9,2*a11^2*a20 + 2*a26*b10*c10 - a20*a28,2*a3*b2*c9 - a17*b1,2*a21*c12^2 + 2*c13*c15^2 - b10^2,s5*b13*c0*c5 - a24*b1,2*c2*c3*c5 - a7*a8,s5*a13*a22^2 + s5*a11*b11*c11 - a13*a23,2*a10*a2*c2 - c0*c7,2*a6*c0*c1 - a8*b4,s5*a10*a22^2 + s5*a12*b11*c11 - a10*a25,s5*a24*b0*c14 + s5*a25*b1*c14,2*a6*b11*c14 + 2*a7*b15*c14,2*a16*b10*b11 - a24*c12,s5*a10*a12*b7 + s5*b8^2*c7 - a9*b7,s5*a8*b1*b4 - a6*c0,2*a5*b10*c2 - b9*c3,a3*b1*c2 - a17*c9,s5*a22*b0*c11 + s5*a19*b1*c11,2*a16*a3*c3 - c0*c10,a2*b3*c0 - a11*b7,s5*a6*b2*c14 - b3*c12,s5*a12*a19^2 + s5*a10*b11*c11 - a12*a25,2*a16*c12*c9 - a27*b15,2*a11*a2*c3 - c0*c7,2*a1*c0*c8 - c1*c7,s5*b2*c0*c9 - a15*a3,2*a22*a23^2 + 2*a25*a28^2 - a18^2 + 1/25*s5,2*a15*b10*b12 - a26*b15,2*a16*b7*c6 - a10*b9,2*a16*c11*c9 - b13*c14,2*a5*b2*b9 - b10*b3,s5*b1*b10*c3 - a18*a3,2*a19^2*a23 + 2*a20^2*a28 - a11^2 + 1/25*s5,2*a10*b15*c8 - a24*b12,b1^2*b5 + 2*a27*c8^2 - b14^2,2*c13*c2*c3 - a20*b4,s5*a18*b3*c1 - a3*b10,2*a9*b7^2 + 2*b9^2*c6 - b8^2,s5*a17*a24^2 + s5*a18*b12*c12 - a17*a22,s5*a6*b7*c4 - c5*c8,2/5*s5*b13*b4 + 2/5*s5*a27*b5 + 1/5*s5*a8*b5,s5*b2*b9*c3 - a5*b10,s5*a24*b1*b5 - b13*c0,2*c12^2*c4 + 2*c15^2*c5 - b3^2,2*a3*b9*c2 - a15*c0,2*a27*c2*c3 - a26*b5,s5*a27*b3*b5 - a26*c2,2*a18*b10*b14 - a28*c15,s5*a20*b12*b2 + s5*a23*b12*b3,s5*a19*b0*b11 + s5*a22*b1*b11,2*a17*b9*c11 - a24*c14,s5*a13*b3*c1 - a2*b8,2*a2*c1*c8 - a12*c2,2*a18*b10*c11 - a23*c12,2*c14^2*c4 + 2*b15^2*c5 - b2^2,2*b1*b14*c0 - a7*b12,2*a12*a15*b6 - b9*c7,b2^2*b5 + 2*b13*b9^2 - b15^2,2*a17^2*a26 + 2*a20*b8*c8 - a25*a26,2*b15^2*c13 + 2*a21*c14^2 - b9^2,2*a12^2*a20 + 2*a26*b9*c9 - a20*a28,s5*a21*b0*b11 + s5*a19*b1*b11,2*a12*b15*c8 - a20*b14,2*a2*b8*c2 - a12*c1,2*a16^2*a26 + 2*a20*b7*c7 - a23*a26,s5*a6*b8*c4 - c5*c7,a2*b3*c1 - a13*b8,s5*a7*b12*c1 - b14*c0,2*a17*b14*b9 - a26*b15,s5*a24*b0*b5 - b13*c1,2*a15*c6*c7 - a12*c9,2*a12*b6*c10 - a11*c9,b2^2*b4 + 2*a21*b9^2 - c14^2,s5*a7*b0*b12 - b1*b14,2*a24*b14*c12 + 2*a26*c15^2,2*a1*b8*c1 - b7*c0,s5*a28*b2*c15 + s5*a26*b3*c15,s5*a12*b1*c2 - a2*c8,2*a20*c2*c3 - b13*b4,s5*a20*b2*b4 - c13*c3,s5*a20*b3*b4 - c13*c2,s5*a6*c5*c8 - b7*c4,2*a2*b7*c3 - a11*c0,s5*a7*b9*c4 - c10*c5,2*a22^2*a25 + 2*a23^2*a28 - a10^2 + 1/25*s5,2*c11^2*c4 + 2*b14^2*c5 - b1^2,s5*a17*b2*c1 - a3*b9,s5*a16*a17^2 + s5*a15*b9*c9 - a14*a16,s5*a11*a19^2 + s5*a13*b11*c11 - a11*a23,2*c10^2*c6 + 2*a9*c8^2 - c7^2,s5*a6*a8*b0 - c1*c4,s5*a8*c2*c5 - a7*b3,s5*a6*c12*c2 - c14*c3,s5*a20*c13*c2 + s5*c12^2*c3 - b3*b4,a6*b1*c1 + 2*a24*b8*c8 - a19*a22,s5*a10^2*a15 + s5*a12*b9*c7 - a15*b6,2*a19*b12*c12 + 2*a24*b15*c15 - a16*a18,s5*a10*a24^2 + s5*a12*b14*c14 - a13*a26,s5*b11^2*c0 + s5*a19*a21*c1 - b0*b4,s5*a10*a22*c14 + s5*c11*c13*c7 - b11*b7,s5*b15^2*c2 + s5*a26*a27*c3 - b2*b5,a5*b2*c2 + 2*a14*b9*c9 - a15*a16,s5*a11*b13*c13 + s5*a13*b14*c14 - a12*a26,2*a19*a25*a6 + 2*a20*a28*a7 + 1/25*s5,2*a22*a24*a6 + 2*a25*a26*a7 + 1/25*s5,s5*c10*c15^2 + s5*a26*a27*c9 - b10*b13,s5*a11*a20*a28 + s5*b12*b15*b7 - a11*a20,2*a22*b12*c12 + 2*a25*b15*c15 - a17*a18,s5*a26*a28*b9 + s5*b10*c15^2 - a20*b9,a7*a8^2 + 2*a20*b4*c4 + 2*a26*b5*c5,s5*a13*a25*c15 + s5*a27*c14*c8 - b14*b8,2*a19*b10*c10 + a6*b3*c3 - a23*a24,2*a23*b10*c10 + 2*a22*b8*c8 - a19*a24,s5*a19*a22*b1 + s5*b0*b11^2 - a6*b1,s5*a16*b13*c13 + s5*a15*b14*c14 - a17*a19,s5*a12*a21^2 + s5*a10*b11*c11 - a11*a20,s5*a11*a18*b1 + s5*b0*b10*c7,2/5*s5*a19*a20 + 2/5*s5*a24*a26 + 1/5*s5*a6*a7,2*a22*a24*a6 + 2*a23*a26*a7 + 1/25*s5,s5*a15*a22*a25 + s5*c11*c14*c9 - a15*a25,s5*a13*a17*c3 + s5*b8*c2*c9,s5*a16*a22*a24 + s5*c10*c11*c12 - a16*a24,s5*a24*b13*c0 + s5*b14^2*c1 - b1*b5,s5*a11*a20*b15 + s5*a27*b12*b7 - c12*c7,s5*a12*a15*c0 + s5*b8*c1*c9,2*b8^2*c11 + 2*a17*b10*c12 - b11*c13,2*b14*b8^2 + 2*a17*b10*c15 - a27*c14,s5*b12^2*c0 + s5*a24*b13*c1 - b0*b5,s5*a10*b12*c12 + s5*a12*b13*c13 - a11*a26,a6*b2*c2 + 2*a19*b9*c9 - a24*a25,s5*a19*a21*b7 + s5*b8*c11^2 - c13*c8,s5*a13*a18^2 + s5*a17*b8*c10 - a13*c6,s5*c10*c12^2 + s5*a20*c13*c9 - a21*b10,s5*a16*a24*c11 + s5*a21*c10*c12 - b10*b12,s5*a17*a22*a24 + s5*b11*c14*c9 - a17*a24,s5*a10*a19^2 + s5*a12*b11*c11 - a13*a20,2*a12*b11*c7 + 2*c14*c9^2 - a21*b14,s5*a13*a17*c10 + s5*b8*c9^2 - c6*c8,s5*a19*a22*b0 + s5*b1*c11^2 - a6*b0,2*a11*a12*a9 + 2*b10*c6*c9 - a11*a12,2*a19*b14*c14 + 2*a24*b15*c15 - a15*a17,s5*a13*a22*a23 + s5*b12*b8*c11 - a13*a22,s5*a18*a28*b14 + s5*b10*b13*b15 - c10*c15,s5*a15*a28*b12 + s5*b13*b9*c15 - b15*c9,a0*b1*c0 + 2*a1*b7*c8 - b0*c1,s5*a24*b12*b3 + s5*b13*b2*c14,2*a11*b8*c4 + 2*a17*c10*c5,2*a15*a7*c10 + 2*a10*a6*c8,2*c12*c7^2 + 2*a16*c15*c9 - a27*b12,s5*a13*a24^2 + s5*a11*b12*c12 - a10*a26,s5*a24*b14*b2 + s5*b13*b3*c12,2*a19*a23*a6 + 2*a20*a28*a7 + 1/25*s5,s5*a11*a21^2 + s5*a13*b11*c11 - a12*a20,a7*b2*c2 + 2*a20*b9*c9 - a26*a28,s5*a16*a23*a26 + s5*c10*c12*c15 - a16*a26,s5*a10*a15^2 + s5*a16*b9*c7 - a10*c6,2*a11*a12*c6 + 2*a14*b10*c9 - b9*c10,a7*b1*c1 + 2*a26*b8*c8 - a20*a25,s5*a26*a27*c2 + s5*c15^2*c3 - b3*b5,a1*b1*c0 + 2*a9*b7*c8 - b8*c7,2*a22*b14*c14 + 2*a23*b15*c15 - a15*a16,2*a12*b12*c7 + 2*b15*c9^2 - b13*c15,a6*a8^2 + 2*a19*b4*c4 + 2*a24*b5*c5,2*a23*b7*c7 + 2*a28*b9*c9 - a20*a26,s5*a11*a19*b12 + s5*b11*b13*b7 - c11*c7,s5*a12*a19*c14 + s5*b11*c13*c8 - b8*c11,s5*a15*a26*b15 + s5*a16*a26*c15 - b10*b12,2*a22*b7*c7 + 2*a25*b9*c9 - a19*a24,s5*a10*a19*b11 + s5*a12*a19*c11 - c14*c8,s5*a15*a23*a28 + s5*b15*c12*c9 - a15*a28,s5*a12*a20*a28 + s5*b14*b8*c15 - a12*a20,2*a13*a6*c7 + 2*a18*a7*c9,s5*a10*a16*b9 + s5*b10^2*c7 - b7*c6,s5*a21*b14*b9 + s5*a17*a24*c11 - c14*c9,2*a20*b10*c10 + a7*b3*c3 - a26*a28,2*a10*a13*a6 + 2*a15*a18*a7 - a6*a7,s5*a18*a22*a23 + s5*b10*b12*c11 - a18*a23,2*a19*a21^2 + 2*a24*b13*c13 + a6*b4*c4,s5*a17*a27^2 + s5*a18*b15*c15 - a16*a24,s5*a15*a25*b11 + s5*a21*b14*b9 - c14*c9,s5*b11^2*b7 + s5*a19*a21*b8 - c13*c7,s5*a12*a20*b15 + s5*a27*c14*c8 - b14*b8,s5*a12*b2*c10 + s5*a18*b3*c8,s5*a15*b7*c2 + s5*a11*b9*c3,2*b11*c14*c4 + 2*b12*b15*c5,2*a16*a17*c6 + 2*a9*b8*c7 - b7*c8,s5*a16*a27^2 + s5*a15*b15*c15 - a17*a24,2*a26*a27^2 + 2*a20*b13*c13 + a7*b5*c5,s5*a17*a26*b14 + s5*b13*b9*c15 - b15*c9,s5*a17*a25*a26 + s5*b15*c14*c9 - a17*a26,s5*a17*b1*b7 + s5*a10*b0*b9,2*a20*b11*c11 + 2*a26*b12*c12 - a10*a11,2*a14*a16*a17 + 2*b8*c6*c7 - a16*a17,a5*b3*c2 + 2*a14*b10*c9 - b9*c10,a1*b0*c0 + 2*a9*b7*c7 - a10*a12,s5*a26*a28*b10 + s5*b15^2*b9 - a20*b10,s5*a13*c1*c10 + s5*a16*c0*c8,2*c10^2*c12 + 2*a11*c11*c8 - a21*b12,2*a14*b10*c10 + a5*b3*c3 - a17*a18,s5*a26*a27*c10 + s5*b15^2*c9 - b13*b9,s5*a19*a21*c0 + s5*c1*c11^2 - b1*b4,s5*a17*a26*b15 + s5*a18*a26*c15 - b14*b9,2*a20*b11*c11 + 2*a26*b14*c14 - a12*a13,s5*a13^2*a18 + s5*a11*b8*c10 - a18*b6,s5*a18*a26^2 + s5*a17*b15*c15 - a15*a24,a1*b1*c1 + 2*a9*b8*c8 - a11*a13,2*a28*b10*c10 + 2*a25*b8*c8 - a20*a26,s5*a18*b12*c12 + s5*a17*b13*c13 - a16*a19,s5*a12*a15*c7 + s5*b9*c8^2 - b6*c9,s5*a13*a19^2 + s5*a11*b11*c11 - a10*a20,s5*a18*a25*a28 + s5*b10*b14*b15 - a18*a28,s5*a13*a22*c12 + s5*b11*c13*c8 - b8*c11,s5*a15*a26^2 + s5*a16*b15*c15 - a18*a24,2*a23*b11*c11 + 2*a28*b14*c14 - a11*a13,s5*b0*b12*c13 + s5*a20*b1*c14,s5*a20*c10*c13 + s5*c14^2*c9 - a21*b9,s5*a20*b0*c12 + s5*b1*b14*c13,s5*a19*a22*c7 + s5*c11^2*c8 - a24*c7,s5*a16*a26*c12 + s5*c10*c13*c15 - b10*b15,s5*a10*a22*a25 + s5*c11*c14*c7 - a10*a22,s5*a26*a28*b2 + s5*b3*c15^2 - a7*b2,a6*b0*c0 + 2*a24*b7*c7 - a19*a22,s5*a11*a19*b11 + s5*a13*a19*c11 - c12*c7,2*c11*c12*c4 + 2*b14*c15*c5,2*a10*a13*a19 + 2*a15*a18*a20 - a19*a20,s5*a11*a18*b8 + s5*b7^2*c10 - b10*b6,s5*a24*b13*b7 + s5*b14^2*b8 - a27*c8,a4*b3*c2 + 2*a5*b10*c9 - b2*c3,s5*a21*b10*b12 + s5*a18*a23*c11 - c10*c12,2*c10^2*c15 + 2*a11*b14*c8 - b13*b15,2*a12*b7*c4 + 2*a16*c5*c9,s5*a15*a20^2 + s5*a16*b12*c12 - a18*a19,s5*a10*a16*b2 + s5*b10*b3*c7,s5*a13*a25*a28 + s5*b14*b15*b8 - a13*a25,s5*c14^2*c2 + s5*a20*c13*c3 - b2*b4,2*a25*b11*c11 + 2*a28*b12*c12 - a10*a12,2*a10*a13*a24 + 2*a15*a18*a26 - a24*a26,s5*b15^2*b2 + s5*a26*a28*b3 - a7*b3,s5*b11^2*c7 + s5*a19*a22*c8 - a24*c8,2*c11*c7^2 + 2*a16*b14*c9 - b11*b13,s5*a11*a19*a23 + s5*b11*b12*b7 - a11*a19,s5*a18*a20^2 + s5*a17*b14*c14 - a15*a19,s5*a10*a23*a28 + s5*b15*c12*c7 - a10*a23,s5*a12*a19*a25 + s5*b14*b8*c11 - a12*a19,a7*b0*c0 + 2*a26*b7*c7 - a20*a23,s5*b12^2*b7 + s5*a24*b13*b8 - a27*c7,s5*a10*a23*b15 + s5*a27*c12*c7 - b12*b7,s5*a19*a24*b10 + s5*a18*b11*c12 - a23*b10,s5*a16*a19*b12 + s5*a23*b11*c10 - a18*b12,2*a16*c10*c12 + 2*b7*c11*c8 - a24*b11,s5*a10*b11*b14 + s5*a19*a24*c7 - a22*c7,2*a10*a12*b12 + 2*a15*b15*c9 - a28*b12,s5*a10*a24*a26 + s5*c14*c15*c7 - a13*a24,s5*a23*a24*b10 + s5*b12*b9*c14 - a19*b10,2*a12*a13*b14 + 2*a18*c15*c9 - a26*b14,s5*a10*a22*b11 + s5*a12*a21*c11 - c14*c7,s5*a24*a25*b2 + s5*b14*b3*c12 - a6*b2,s5*a28*b10*b15 + s5*a26*b15*b9 - a18*c14,s5*a10*b12*c15 + s5*a20*a26*c7 - a23*c7,2*a16*a18*b15 + 2*a13*c14*c7 - a24*b15,s5*a22*a24*b7 + s5*a11*c11*c12 - a19*b7,s5*a12*a21*b14 + s5*a24*c11*c8 - b11*c7,s5*a20*b10*b14 + s5*a25*b14*b9 - a15*c11,s5*a15*a19*a20 + s5*b11*c12*c9 - a18*a20,s5*a10*a19*a20 + s5*b11*c12*c7 - a13*a19,s5*a12*a20*c11 + s5*a22*c12*c8 - a13*c11,s5*a20*b9*c11 + s5*a17*a22*c12 - a18*c12,s5*a18*a19*a20 + s5*b10*b11*b14 - a15*a20,2*a19*a24*a25 + 2*a20*a26*a28 - b9*c9 + 1/25*s5,2*a12*a9*c7 + 2*a16*c6*c9 - a10*c7,2*a13*b8*c11 + 2*a17*a18*c12 - a22*c12,s5*a16*b14*c13 + s5*a15*a25*c14 - b11*b9,2*a7*b15^2 + 2*a6*b12*c14 - b3*c2,2*a16*a18*b12 + 2*a13*b11*c7 - a19*b12,s5*a15*a16*b2 + s5*b10*b3*c9 - a5*b2,2*a24*c11^2 + 2*a26*b14*c12 - b8*c7,2*a21*b7*b8 + 2*a16*a17*c13 - a19*c13,s5*a11*a13*b1 + s5*b0*b8*c7 - a1*b1,2*a11*b12*b7 + 2*b15*b9*c10 - a20*c15,s5*a20*a25*b1 + s5*b0*b12*c14 - a7*b1,2*a10*a9*b7 + 2*a15*b9*c6 - a12*b7,s5*a22*a24*b8 + s5*a12*b11*c14 - a19*b8,s5*a15*a28*b15 + s5*a16*a27*c15 - b12*b9,s5*a16*a27*b12 + s5*a20*b15*c10 - c15*c9,s5*a19*c11*c7 + s5*a21*c11*c8 - a11*b12,s5*a16*b8*c7 + s5*a13*b10*c8 - a17*b6,s5*a12*a25*b11 + s5*a19*c14*c8 - a10*b11,2*a24*b11*c14 + 2*a26*b15*c14 - a17*b9,s5*a12*a20*a21 + s5*c11*c12*c8 - a11*a21,s5*a22*c11*c7 + s5*a19*c11*c8 - a10*b14,s5*a11*a20*a21 + s5*b11*c14*c7 - a12*a21,2*a19*a22*a24 + 2*a20*a25*a26 - b8*c8 + 1/25*s5,s5*a18*a22*c12 + s5*b11*c10*c13 - a17*c12,s5*a17*a19*c13 + s5*b11*b9*c14 - a16*c13,s5*a26*b10*b12 + s5*a18*a24*c15 - a16*c15,s5*a24*b7*c11 + s5*a11*a21*c12 - b11*b8,s5*a17*b8*c7 + s5*a10*b7*c9 - a16*b6,2*a16*a17*a27 + 2*b13*b7*b8 - a24*a27,s5*a11*a26*c12 + s5*a23*b7*c15 - a10*c12,2*a11*a9*b8 + 2*a17*b10*c6 - a13*b8,s5*a10*a12*b0 + s5*b1*b7*c8 - a1*b0,s5*a13*a20*c11 + s5*a19*c14*c8 - a12*c11,2*a10*a12*b11 + 2*a15*c14*c9 - a25*b11,s5*a24*b10*b11 + s5*a18*a19*c12 - a16*c12,2*a14*a17*b10 + 2*a11*b8*c6 - a18*b10,s5*a17*a24*a27 + s5*b12*b15*b9 - a16*a27,s5*a20*b15*b9 + s5*a17*a27*c14 - b10*c15,2*a19*c11*c12 + 2*a20*c12*c15 - a11*b7,s5*a12*a26*b13 + s5*b14*c15*c8 - a11*b13,s5*a11*a23*b11 + s5*a19*b12*b7 - a13*b11,2*a10*a11*b12 + 2*a15*b15*c10 - a26*b12,s5*a15*b0*b9 + s5*a17*b1*b9 - a3*b2,s5*a21*b12*b8 + s5*a13*a23*c11 - a11*c11,s5*a17*b14*c11 + s5*a19*a25*c9 - a24*c9,s5*a24*a25*b9 + s5*b10*b14*c12 - a19*b9,s5*a18*a23*c12 + s5*a17*b12*c13 - b10*c11,s5*a12*b9*c10 + s5*a18*b10*c8 - a11*c6,s5*a12*a26*b14 + s5*a25*c15*c8 - a13*b14,2*a19*a22*a6 + 2*a20*a25*a7 - b1*c1 + 1/25*s5,2*a19*a23*a24 + 2*a20*a26*a28 - b10*c10 + 1/25*s5,2*a11*a13*b14 + 2*a18*c10*c15 - a28*b14,s5*a23*b10*b12 + s5*a20*b12*b9 - a18*b11,s5*a18*a26*b14 + s5*a20*b15*c10 - c15*c9,s5*a26*b14*b9 + s5*a17*a25*c15 - a18*c15,2*a6*b11^2 + 2*a7*b12*c14 - b1*c0,s5*a16*a24*a27 + s5*b10*b14*c15 - a17*a27,2*a11*a12*c4 + 2*c10*c5*c9 - a7*c4,s5*b10*c11*c12 + s5*a16*a19*c13 - a17*c13,2*a24*a25*a6 + 2*a26*a28*a7 - b2*c2 + 1/25*s5,2*a20*b15^2 + 2*a19*b12*c14 - b10*c9,s5*a15*a23*b15 + s5*a27*c12*c9 - a16*b15,s5*a24*b7*c11 + s5*a10*a19*c14 - b11*b8,s5*a25*a26*b8 + s5*a12*b15*c14 - a20*b8,s5*a25*b11*b9 + s5*a17*a19*c14 - a15*c14,s5*a21*b11*c7 + s5*a19*b11*c8 - a12*b14,2*a15*a16*b15 + 2*a10*b12*b7 - a23*b15,s5*a15*a24*a26 + s5*c14*c15*c9 - a18*a26,s5*a23*a24*b3 + s5*b12*b2*c14 - a6*b3,s5*a17*a18*b3 + s5*b2*b9*c10 - a5*b3,s5*a15*a22*c14 + s5*c11*c13*c9 - a16*c14,s5*a23*c12*c7 + s5*a24*c12*c8 - a10*c15,s5*a24*b9*c11 + s5*a15*a19*c14 - a17*c14,2*a18*c10*c6 + 2*a13*a9*c8 - a11*c8,s5*a10*b2*c7 + s5*a11*b3*c7 - a2*b0,s5*a12*a28*b12 + s5*a24*b15*c8 - a10*b12,2*b11*b8*c7 + 2*a17*c14*c9 - a24*c11,s5*a19*b11*c7 + s5*a22*b11*c8 - a13*b12,s5*a13*a28*b14 + s5*b13*b15*b8 - a11*b14,2*a11*a12*a21 + 2*c10*c13*c9 - a20*a21,s5*a16*b11*b12 + s5*a19*a23*c10 - a24*c10,2*a15*a17*b15 + 2*a10*b12*b8 - a24*b15,s5*a13*a19*a20 + s5*b11*b14*b8 - a10*a19,2*a14*a15*b9 + 2*a10*b7*c6 - a16*b9,s5*a16*b12*b15 + s5*a20*a28*c10 - a26*c10,s5*a24*b15*b7 + s5*a11*a28*c14 - a13*c14,s5*a24*c14*c7 + s5*a25*c14*c8 - a13*b15,2*a14*a18*c10 + 2*a13*c6*c8 - a17*c10,s5*a11*a26*b13 + s5*b12*b15*c7 - a12*b13,2*a12*a13*c11 + 2*a18*c12*c9 - a20*c11,s5*a26*b10*c15 + s5*a28*b9*c15 - a15*c12,s5*a20*a23*b0 + s5*b1*b14*c12 - a7*b0,s5*a17*a24*b15 + s5*a28*b12*b9 - a15*b15,s5*a16*a24*b15 + s5*a28*c10*c14 - a18*b15,s5*a13*a25*b14 + s5*a11*b13*c14 - c15*c8,s5*a11*a21*b11 + s5*a13*a22*c11 - c12*c8,s5*a20*a26*b8 + s5*a13*c14*c15 - a25*b8,2*a12*b14*b8 + 2*b10*c15*c9 - a20*b15,2*a19*a22*a24 + 2*a20*a23*a26 - b7*c7 + 1/25*s5,s5*a27*c10*c14 + s5*a18*a25*c15 - a17*c15,s5*a16*b0*b10 + s5*a18*b1*b10 - a3*b3,2*a11*a12*b13 + 2*a27*c10*c9 - a26*b13,s5*a17*b14*c15 + s5*a20*a28*c9 - a26*c9,s5*b12*c14*c7 + s5*a20*a25*c8 - a26*c8,2*a24*c11*c12 + 2*a26*c12*c15 - a16*b10,s5*a11*b9*c10 + s5*a15*b7*c9 - a12*c6,s5*a15*a24*b15 + s5*a26*b14*b9 - a17*b15,2*a10*a11*b11 + 2*a15*c10*c14 - a20*b11,s5*a17*a27*b15 + s5*a18*a28*c15 - b10*b14,2*a19*b11*c14 + 2*a20*b15*c14 - a12*b8,a6*b1*c0 + 2*a24*b7*c8 - b11^2,s5*a10*a20*b11 + s5*a19*c12*c7 - a11*b11,s5*a27*b10*c15 + s5*a26*b9*c15 - a17*c14,2*a10*b11*b8 + 2*a15*a17*c14 - a19*c14,2*a23*a24*a6 + 2*a26*a28*a7 - b3*c3 + 1/25*s5,s5*a13*a26*b14 + s5*a20*b15*c8 - a12*b14,2*a10*b11*b7 + 2*a15*a16*c14 - a22*c14,s5*a20*a23*c7 + s5*b14*c12*c8 - a26*c7,s5*a15*b11*b14 + s5*a19*a24*c9 - a25*c9,s5*a26*c10*c12 + s5*a16*a23*c15 - a15*c15,s5*a24*b11*b8 + s5*a13*a19*c12 - b7*c11,s5*a10*a26*b12 + s5*a20*c15*c7 - a11*b12,s5*a22*b14*b7 + s5*a11*a20*c11 - a10*c11,s5*a20*a26*b10 + s5*a18*c14*c15 - a28*b10,s5*a15*a26*b12 + s5*a20*c15*c9 - b15*c10,s5*a13*a24*a26 + s5*b12*b8*c15 - a10*a24,s5*a16*a22*b14 + s5*a20*c10*c11 - a15*b14,2*a12*c6*c7 + 2*a14*a16*c9 - a15*c9,s5*a12*b2*c8 + s5*a13*b3*c8 - a2*b1,s5*a19*a24*b8 + s5*a13*b11*c12 - a22*b8,s5*a10*a23*b12 + s5*a12*b13*c12 - b15*c7,2*a11*a13*c11 + 2*a18*c10*c12 - a23*c11,a7*b3*c2 + 2*a20*b10*c9 - b15^2,s5*a15*b12*c15 + s5*a20*a26*c9 - a28*c9,s5*a23*a26*b7 + s5*a11*c12*c15 - a20*b7,s5*a10*a25*b11 + s5*a21*b14*b7 - a12*b11,s5*a10*a28*b12 + s5*b13*b7*c15 - a12*b12,2*a19*a22*a6 + 2*a20*a23*a7 - b0*c0 + 1/25*s5,2*b7*b8*c4 + 2*a16*a17*c5 - a6*c5,s5*a26*b10*b15 + s5*a27*b15*b9 - a16*c12,2*a13*b14*b8 + 2*a17*a18*c15 - a25*c15,s5*a18*a24*a26 + s5*b10*b12*c15 - a15*a26,s5*a20*a21*b9 + s5*a17*c11*c12 - c10*c13,2*a16*c10*c15 + 2*b14*b7*c8 - a26*b12,s5*a27*b10*b13 + s5*a18*b14*b15 - a26*c9,s5*a12*b11*c14 + s5*a19*c13*c8 - a21*b7,s5*a12*b13*c15 + s5*a26*b14*c8 - b12*c7,s5*a13*b14*b15 + s5*a27*b13*b8 - a24*c7,2*a11*b13*b8 + 2*a17*a27*c10 - b14*c15,2*a11*b11*b7 + 2*b9*c10*c14 - a19*c12,a6*b3*c2 + 2*a19*b10*c9 - b12*c14,s5*a10*c11*c14 + s5*a21*c13*c7 - a19*b8,2*b12*b15*c13 + 2*a21*b11*c14 - a12*b7,b0*b1*b4 + 2*c13*c7*c8 - a19*a21,a8*b2*b3 + 2*b10*b9*c4 - a7*c5,2*a19*c13*c4 + 2*a24*a27*c5 + a6*a8*c5,2*b14*c11*c13 + 2*a27*c12*c15 - a16*b9,2*a18*a23*c10 + 2*a13*a22*c8 - b12*c11,2*a18*a28*c10 + 2*a13*a25*c8 - b14*b15,s5*a11*c11*c12 + s5*a19*c13*c7 - a21*b8,2*a16*a24*c10 + 2*a11*a19*c7 - b11*b12,2*a20*a21^2 + 2*a26*b13*c13 + a7*b4*c4 - a11*a12,2*b12*b8*c7 + 2*a17*b15*c9 - a26*b14,2*a23*b11*c12 + 2*a28*c14*c15 - a18*b10,2*a16*a3*c10 + 2*a11*a2*c7 - b0*c3,2*a16*a26*c10 + 2*a11*a20*c7 - b12*b15,2*a28*b15*c12 + 2*a25*c11*c14 - a15*b9,2*a11*a21*b8 + 2*a17*c10*c13 - c11*c12,2*a15*a26*c10 + 2*a10*a24*c8 - b12*c15,s5*a10*b15*c12 + s5*a27*c13*c7 - a24*b8,s5*a12*b12*b15 + s5*a24*a27*c8 - b13*b7,2*a23*b15*c12 + 2*a22*c11*c14 - a10*b7,s5*a11*b14*c15 + s5*a24*a27*c7 - b13*b8,2*a12*b13*b7 + 2*a16*a27*c9 - b12*b15,s5*a11*a20*b12 + s5*a13*a24*b14 - c15*c7,s5*a15*c11*c14 + s5*a21*c13*c9 - a20*b10,s5 ...
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answered 2020-08-11 11:18:19 -0600

rburing gravatar image

updated 2020-08-12 03:54:26 -0600

This output should never occur, because Id.normal_basis() should return a list/sequence of monomials which form a vector space basis of the quotient ring, and 0 is not a monomial (it is the "empty sum of monomials").

If Id.dimension() > 0 then the quotient is not finite-dimensional (i.e. the normal basis is infinite), and the convention seems to be for normal_basis() to return the empty list [] in this case (it does not mean that the normal basis is actually empty, it just means we cannot really return the whole thing in a reasonable way). In recent versions of SageMath you can still obtain part of the (infinite) basis, by passing a degree argument to normal_basis; this will return all monomials of the specified degree in the basis:

sage: R.<x,y,z>=PolynomialRing(QQ)
sage: I=R.ideal([x^2+y^2-1, z])
sage: I.dimension()
1
sage: I.normal_basis(1)
[y, x]
sage: I.normal_basis(2)
[y^2, x*y]

In the above output z is absent because $z=0$ in the quotient, and x^2 is absent because $x^2 = 1 - y^2$ in the quotient, etc.

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I edited all the details of the computation, so you can do it yourself (30 min on my laptop). According to SageMath, it is 2-codim ideal of normal-basis [0]. Is it a bug? Should I understand that the dim in in fact (-1)?

Sébastien Palcoux gravatar imageSébastien Palcoux ( 2020-08-12 00:36:56 -0600 )edit

You have indeed found a bug in Singular and/or SageMath. According to the documentation of Singular's kbase it should return -1 when the quotient is not finite-dimensional, but here it returns [0]. In SageMath's normal_basis() there was code added in trac ticket #29543 that should filter out the zero, so that ticket could be a natural place to bring this up. In any case, $0$ can never be an element of any vector space basis, and if an ideal is positive-dimensional then the quotient ring has an infinite basis as a vector space, which can be accessed degree-wise as described in the answer.

rburing gravatar imagerburing ( 2020-08-12 03:46:31 -0600 )edit

There is a misunderstanding: the ideal is not finite-dimensional, but finite co-dimensional, the function Id.dimension() is confusing as it provides the dimension of the ring modulo the ideal, i.e. the dimension of the quotient. So the quotient is 2-dim (this is what I mean by "the ideal is 2-codim").

Sébastien Palcoux gravatar imageSébastien Palcoux ( 2020-08-12 04:06:52 -0600 )edit

By the dimension of an ideal $J$, I (and Sage and Singular) mean the Krull dimension of the ring modulo $J$. Yes, the Krull dimension of the quotient is 2, but (hence) the vector space dimension of the quotient is $\infty$. The normal basis is a basis of the quotient as a vector space, so it is infinite. A "witness" of the fact that the Krull dimension of the quotient is (at least) 2 would be: a chain of prime ideals of length 2 in the quotient.

rburing gravatar imagerburing ( 2020-08-12 04:27:38 -0600 )edit

In the example in the answer, the vanishing locus $V(I)$ of $I$ is the intersection of the cylinder $x^2+y^2=1$ and the plane $z=0$, so it is just a circle in the plane, i.e. a 1-dimensional algebraic variety (and this dimension is the same as the Krull dimension of the quotient ring). The quotient ring can be identified with the ring of polynomial functions on $V(I)$, i.e. the set of polynomial functions on the circle. On the one hand the circle is 1-dimensional (Krull dimension of the quotient), but on the other hand the functions $1, y, y^2, y^3 \ldots$ on the circle are all linearly independent because there is no equation that could relate them (they are part of an infinite normal basis, so the vector space dimension of the quotient is infinite).

rburing gravatar imagerburing ( 2020-08-12 04:43:43 -0600 )edit

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Asked: 2020-08-11 07:22:14 -0600

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