The realization space is [1 1 0 0 1 1 0 x1^2*x2*x3 - x1^2*x2 - x1^2*x3 + x1^2 - 2*x1*x2*x3^2 + 2*x1*x2*x3 + 2*x1*x3^2 - 2*x1*x3 + x2*x3^3 - x2*x3^2 - x3^3 + x3^2 x1^2*x2*x3 - x1^2*x2 - x1^2*x3 + x1^2 - 2*x1*x2*x3^2 + 2*x1*x2*x3 + 2*x1*x3^2 - 2*x1*x3 + x2*x3^3 - x2*x3^2 - x3^3 + x3^2 1 1] [1 0 1 0 1 0 x1*x2 - x1 - x2*x3 + x3 x1^2*x2^2*x3 - x1^2*x2^2 - x1^2*x2*x3 + x1^2*x2 - 3*x1*x2^2*x3^2 + 2*x1*x2^2*x3 + x1*x2*x3^3 + 3*x1*x2*x3^2 - 2*x1*x2*x3 - x1*x3^3 + x2^2*x3^3 - 2*x2*x3^3 + x3^3 x1^2*x2^2*x3 - x1^2*x2^2 - x1^2*x2*x3 + x1^2*x2 - 3*x1*x2^2*x3^2 + 2*x1*x2^2*x3 + x1*x2*x3^3 + 3*x1*x2*x3^2 - 2*x1*x2*x3 - x1*x3^3 + x2^2*x3^3 - 2*x2*x3^3 + x3^3 x1 x2] [0 0 0 1 1 1 x2*x3 - x3^2 x1^2*x2*x3^2 - x1^2*x2*x3 - x1^2*x3^2 + x1^2*x3 - 3*x1*x2*x3^3 + 3*x1*x2*x3^2 + x1*x3^4 + x1*x3^3 - 2*x1*x3^2 + x2*x3^4 - x2*x3^2 - 2*x3^4 + 2*x3^3 x1^2*x2*x3^2 - x1^2*x2*x3 - x1^2*x3^2 + x1^2*x3 - 3*x1*x2*x3^3 + 4*x1*x2*x3^2 - x1*x2*x3 + x1*x3^4 - x1*x3^2 + x2*x3^4 - x2*x3^3 - x3^4 + x3^3 x3 x3] in the multivariate polynomial ring in 3 variables over ZZ within the vanishing set of the ideal Ideal with 3 generators avoiding the zero loci of the polynomials RingElem[x1 - x2, x3, x3 - 1, x1^2*x2 - x1^2 - 3*x1*x2*x3 + x1*x3^2 + 2*x1*x3 + x2*x3^2 - x3^2, x1^3*x2 - x1^3 - x1^2*x2^2 - 3*x1^2*x2*x3 + x1^2*x2 + x1^2*x3^2 + 2*x1^2*x3 + 2*x1*x2^2*x3 + x1*x2*x3^2 - x1*x2*x3 - 2*x1*x3^2 - x2^2*x3^2 + x2*x3^2, x1, x2 - x3, x1^3*x2 - x1^3 - x1^2*x2^2 - 3*x1^2*x2*x3 + x1^2*x2 + x1^2*x3^2 + 2*x1^2*x3 + 3*x1*x2^2*x3 - x1*x2^2 - x1*x2*x3 + x1*x2 - x1*x3^2 - x1*x3 - x2^2*x3^2 + x2*x3^2, x1^3*x2 - x1^3 - x1^2*x2^2 - 3*x1^2*x2*x3 + x1^2*x2 + x1^2*x3^2 + 2*x1^2*x3 + 3*x1*x2^2*x3 - x1*x2^2 - 2*x1*x3^2 - x2^2*x3^2 + x2*x3^2, x1^3*x2 - x1^3 - x1^2*x2^2 - 3*x1^2*x2*x3 + x1^2*x2 + x1^2*x3^2 + 2*x1^2*x3 + 3*x1*x2^2*x3 - x1*x2*x3 - 2*x1*x3^2 - x2^2*x3^2 + x2*x3^2, x1^6*x2^2 - 2*x1^6*x2 + x1^6 - 2*x1^5*x2^3 - 6*x1^5*x2^2*x3 + 4*x1^5*x2^2 + 2*x1^5*x2*x3^2 + 10*x1^5*x2*x3 - 2*x1^5*x2 - 2*x1^5*x3^2 - 4*x1^5*x3 + x1^4*x2^4 + 11*x1^4*x2^3*x3 - 2*x1^4*x2^3 + 8*x1^4*x2^2*x3^2 - 18*x1^4*x2^2*x3 + x1^4*x2^2 - 6*x1^4*x2*x3^3 - 14*x1^4*x2*x3^2 + 7*x1^4*x2*x3 + x1^4*x3^4 + 4*x1^4*x3^3 + 7*x1^4*x3^2 - 5*x1^3*x2^4*x3 - 18*x1^3*x2^3*x3^2 + 7*x1^3*x2^3*x3 + 2*x1^3*x2^2*x3^3 + 28*x1^3*x2^2*x3^2 - x1^3*x2^2*x3 + x1^3*x2*x3^4 + 8*x1^3*x2*x3^3 - 13*x1^3*x2*x3^2 - x1^3*x2*x3 - 3*x1^3*x3^4 - 6*x1^3*x3^3 + x1^3*x3^2 + 8*x1^2*x2^4*x3^2 + 9*x1^2*x2^3*x3^3 - 9*x1^2*x2^3*x3^2 - 2*x1^2*x2^2*x3^4 - 22*x1^2*x2^2*x3^3 + x1^2*x2*x3^4 + 13*x1^2*x2*x3^3 + 2*x1^2*x2*x3^2 + 2*x1^2*x3^4 - 2*x1^2*x3^3 - 5*x1*x2^4*x3^3 - x1*x2^3*x3^4 + 7*x1*x2^3*x3^3 + 5*x1*x2^2*x3^4 - x1*x2^2*x3^3 - 5*x1*x2*x3^4 - x1*x2*x3^3 + x1*x3^4 + x2^4*x3^4 - 2*x2^3*x3^4 + x2^2*x3^4, x1^3*x2 - x1^3 - x1^2*x2^2 - 3*x1^2*x2*x3 + 2*x1^2*x2 + x1^2*x3^2 + 2*x1^2*x3 - x1^2 + 3*x1*x2^2*x3 - 4*x1*x2*x3 - x1*x3^2 + 2*x1*x3 - x2^2*x3^2 + 2*x2*x3^2 - x3^2, x1^3*x2*x3 - x1^3*x2 - x1^3*x3 + x1^3 - x1^2*x2^2*x3 + x1^2*x2^2 - 3*x1^2*x2*x3^2 + 5*x1^2*x2*x3 - x1^2*x2 + x1^2*x3^3 + x1^2*x3^2 - 3*x1^2*x3 + 3*x1*x2^2*x3^2 - 3*x1*x2^2*x3 - 4*x1*x2*x3^2 + 2*x1*x2*x3 - x1*x3^3 + 3*x1*x3^2 - x2^2*x3^3 + x2^2*x3^2 + 2*x2*x3^3 - x2*x3^2 - x3^3, x1^3*x2^2*x3 - x1^3*x2^2 - x1^3*x2*x3 + x1^3*x2 - x1^2*x2^3*x3 + x1^2*x2^3 - 3*x1^2*x2^2*x3^2 + 5*x1^2*x2^2*x3 - x1^2*x2^2 + x1^2*x2*x3^3 + x1^2*x2*x3^2 - 4*x1^2*x2*x3 + x1^2*x3 + 3*x1*x2^3*x3^2 - 3*x1*x2^3*x3 - 4*x1*x2^2*x3^2 + 2*x1*x2^2*x3 - x1*x2*x3^3 + 5*x1*x2*x3^2 - 2*x1*x3^2 - x2^3*x3^3 + x2^3*x3^2 + 2*x2^2*x3^3 - x2^2*x3^2 - 2*x2*x3^3 + x3^3, x1^3*x2^2 - x1^3*x2 - x1^2*x2^3 - 3*x1^2*x2^2*x3 + 2*x1^2*x2^2 + x1^2*x2*x3^2 + 2*x1^2*x2*x3 - 2*x1^2*x2 + x1^2 + 3*x1*x2^3*x3 - 4*x1*x2^2*x3 - x1*x2*x3^2 + 4*x1*x2*x3 - 2*x1*x3 - x2^3*x3^2 + 2*x2^2*x3^2 - 2*x2*x3^2 + x3^2, x2 + x3 - 1, x2, x2 - 1, x1 - 1, x1 - x3, x1^4*x2 - x1^4 - x1^3*x2^2 - 3*x1^3*x2*x3 + x1^3*x3^2 + 2*x1^3*x3 + x1^3 + 3*x1^2*x2^2*x3 + 2*x1^2*x2^2 + x1^2*x2*x3 - 3*x1^2*x2 - 2*x1^2*x3^2 - 2*x1^2*x3 + x1^2 - x1*x2^2*x3^2 - 5*x1*x2^2*x3 + x1*x2*x3^2 + 6*x1*x2*x3 + x1*x3^2 - 2*x1*x3 + 2*x2^2*x3^2 - 3*x2*x3^2 + x3^2, x1^3*x2*x3 - x1^3*x2 - x1^3*x3 + x1^3 - x1^2*x2^2*x3 + x1^2*x2^2 - 3*x1^2*x2*x3^2 + 4*x1^2*x2*x3 - x1^2*x2 + x1^2*x3^3 + x1^2*x3^2 - 2*x1^2*x3 + 3*x1*x2^2*x3^2 - 2*x1*x2^2*x3 - 3*x1*x2*x3^2 + x1*x2*x3 - x1*x3^3 + 2*x1*x3^2 - x2^2*x3^3 + x2^2*x3^2 + x2*x3^3 - x2*x3^2, x1^3*x2 - x1^3 - x1^2*x2^2 - 3*x1^2*x2*x3 + x1^2*x2 + x1^2*x3^2 + 2*x1^2*x3 + 3*x1*x2^2*x3 - 2*x1*x2*x3 - x1*x3^2 - x2^2*x3^2 - x2^2*x3 + 2*x2*x3^2 + x2*x3 - x3^2, x1^5*x2^2*x3 - x1^5*x2^2 - x1^5*x2*x3^2 - x1^5*x2*x3 + 2*x1^5*x2 + x1^5*x3^2 - x1^5 - x1^4*x2^3*x3 + x1^4*x2^3 - 2*x1^4*x2^2*x3^2 + 5*x1^4*x2^2*x3 - 2*x1^4*x2^2 + 4*x1^4*x2*x3^3 + x1^4*x2*x3^2 - 8*x1^4*x2*x3 + x1^4*x2 - x1^4*x3^4 - 2*x1^4*x3^3 + 4*x1^4*x3 + 3*x1^3*x2^3*x3^2 - 3*x1^3*x2^3*x3 - 3*x1^3*x2^2*x3^3 - 8*x1^3*x2^2*x3^2 + 6*x1^3*x2^2*x3 - x1^3*x2*x3^3 + 14*x1^3*x2*x3^2 - 3*x1^3*x2*x3 + 2*x1^3*x3^4 - 7*x1^3*x3^2 - x1^2*x2^3*x3^3 + 3*x1^2*x2^3*x3^2 + x1^2*x2^2*x3^4 + 9*x1^2*x2^2*x3^3 - 8*x1^2*x2^2*x3^2 - 2*x1^2*x2*x3^4 - 12*x1^2*x2*x3^3 + 4*x1^2*x2*x3^2 + 6*x1^2*x3^3 - 3*x1*x2^3*x3^3 - 2*x1*x2^2*x3^4 + 6*x1*x2^2*x3^3 + 4*x1*x2*x3^4 - 3*x1*x2*x3^3 - 2*x1*x3^4 + x2^3*x3^4 - 2*x2^2*x3^4 + x2*x3^4, x1^4*x2*x3 - x1^4*x2 - x1^4*x3 + x1^4 - x1^3*x2^2*x3 + x1^3*x2^2 - 3*x1^3*x2*x3^2 + 3*x1^3*x2*x3 + x1^3*x3^3 + x1^3*x3^2 - x1^3*x3 - x1^3 + 3*x1^2*x2^2*x3^2 - x1^2*x2^2*x3 - x1^2*x2^2 + x1^2*x2*x3^2 - 4*x1^2*x2*x3 + x1^2*x2 - 2*x1^2*x3^3 + 3*x1^2*x3 - x1*x2^2*x3^3 - 4*x1*x2^2*x3^2 + 3*x1*x2^2*x3 + x1*x2*x3^3 + 5*x1*x2*x3^2 - 2*x1*x2*x3 + x1*x3^3 - 3*x1*x3^2 + 2*x2^2*x3^3 - x2^2*x3^2 - 3*x2*x3^3 + x2*x3^2 + x3^3, x1^4*x2*x3 - x1^4*x2 - x1^4*x3 + x1^4 - x1^3*x2^2*x3 + x1^3*x2^2 - 3*x1^3*x2*x3^2 + 4*x1^3*x2*x3 - x1^3*x2 + x1^3*x3^3 + x1^3*x3^2 - 2*x1^3*x3 + 3*x1^2*x2^2*x3^2 - 2*x1^2*x2^2*x3 - 2*x1^2*x2*x3^2 - x1^2*x3^3 + x1^2*x3^2 + x1^2*x3 - x1*x2^2*x3^3 - x1*x2^2*x3^2 + x1*x2*x3^3 + 3*x1*x2*x3^2 - 2*x1*x3^2 + x2^2*x3^3 - 2*x2*x3^3 + x3^3, x1^4*x2 - x1^4 - x1^3*x2^2 - 3*x1^3*x2*x3 + x1^3*x2 + x1^3*x3^2 + 2*x1^3*x3 + 3*x1^2*x2^2*x3 + x1^2*x2^2 - 2*x1^2*x2*x3 - 2*x1^2*x2 - x1^2*x3^2 + x1^2 - x1*x2^2*x3^2 - 2*x1*x2^2*x3 + x1*x2*x3^2 + 4*x1*x2*x3 - 2*x1*x3 + x2^2*x3^2 - 2*x2*x3^2 + x3^2, x1 + x3 - 1, x1^2*x2 - x1^2 - 3*x1*x2*x3 + x1*x3^2 + 2*x1*x3 + x2*x3^2 + x2*x3 - 2*x3^2, x1^6*x2^2*x3 - x1^6*x2^2 - 2*x1^6*x2*x3 + 2*x1^6*x2 + x1^6*x3 - x1^6 - x1^5*x2^3*x3 + x1^5*x2^3 - 6*x1^5*x2^2*x3^2 + 7*x1^5*x2^2*x3 - x1^5*x2^2 + 2*x1^5*x2*x3^3 + 8*x1^5*x2*x3^2 - 9*x1^5*x2*x3 - x1^5*x2 - 2*x1^5*x3^3 - 2*x1^5*x3^2 + 3*x1^5*x3 + x1^5 + 6*x1^4*x2^3*x3^2 - 4*x1^4*x2^3*x3 - x1^4*x2^3 + 9*x1^4*x2^2*x3^3 - 13*x1^4*x2^2*x3^2 - x1^4*x2^2*x3 + 2*x1^4*x2^2 - 6*x1^4*x2*x3^4 - 10*x1^4*x2*x3^3 + 10*x1^4*x2*x3^2 + 10*x1^4*x2*x3 - x1^4*x2 + x1^4*x3^5 + 3*x1^4*x3^4 + 4*x1^4*x3^3 - 4*x1^4*x3^2 - 5*x1^4*x3 - 11*x1^3*x2^3*x3^3 + 7*x1^3*x2^3*x3 + 8*x1^3*x2^2*x3^3 + 16*x1^3*x2^2*x3^2 - 12*x1^3*x2^2*x3 + x1^3*x2*x3^5 + 11*x1^3*x2*x3^4 - 6*x1^3*x2*x3^3 - 23*x1^3*x2*x3^2 + 5*x1^3*x2*x3 - 3*x1^3*x3^5 - 5*x1^3*x3^4 + 3*x1^3*x3^3 + 9*x1^3*x3^2 + 6*x1^2*x2^3*x3^4 + 13*x1^2*x2^3*x3^3 - 13*x1^2*x2^3*x3^2 - x1^2*x2^2*x3^5 - 11*x1^2*x2^2*x3^4 - 26*x1^2*x2^2*x3^3 + 20*x1^2*x2^2*x3^2 - x1^2*x2*x3^5 + 3*x1^2*x2*x3^4 + 24*x1^2*x2*x3^3 - 8*x1^2*x2*x3^2 + 3*x1^2*x3^5 - x1^2*x3^4 - 8*x1^2*x3^3 - x1*x2^3*x3^5 - 10*x1*x2^3*x3^4 + 7*x1*x2^3*x3^3 + 4*x1*x2^2*x3^5 + 20*x1*x2^2*x3^4 - 12*x1*x2^2*x3^3 - 3*x1*x2*x3^5 - 14*x1*x2*x3^4 + 5*x1*x2*x3^3 + 4*x1*x3^4 + 2*x2^3*x3^5 - x2^3*x3^4 - 5*x2^2*x3^5 + 2*x2^2*x3^4 + 4*x2*x3^5 - x2*x3^4 - x3^5, x1^3*x2 - x1^3 - 2*x1^2*x2*x3 - x1^2*x2 + x1^2*x3^2 + x1^2*x3 + x1^2 - 2*x1*x2*x3^2 + 4*x1*x2*x3 + x1*x3^3 - x1*x3^2 - 2*x1*x3 + x2*x3^3 - x2*x3^2 - x3^3 + x3^2, x1^2*x2 - x1^2 - 3*x1*x2*x3 + x1*x2 + x1*x3^2 + x1*x3 + x2*x3^2 - x3^2, x1^6*x2^2*x3 - x1^6*x2^2 - 2*x1^6*x2*x3 + 2*x1^6*x2 + x1^6*x3 - x1^6 - x1^5*x2^3*x3 + x1^5*x2^3 - 6*x1^5*x2^2*x3^2 + 9*x1^5*x2^2*x3 - 3*x1^5*x2^2 + 2*x1^5*x2*x3^3 + 8*x1^5*x2*x3^2 - 13*x1^5*x2*x3 + 3*x1^5*x2 - 2*x1^5*x3^3 - 2*x1^5*x3^2 + 5*x1^5*x3 - x1^5 + 6*x1^4*x2^3*x3^2 - 6*x1^4*x2^3*x3 + x1^4*x2^3 + 9*x1^4*x2^2*x3^3 - 23*x1^4*x2^2*x3^2 + 13*x1^4*x2^2*x3 - 2*x1^4*x2^2 - 6*x1^4*x2*x3^4 - 8*x1^4*x2*x3^3 + 26*x1^4*x2*x3^2 - 10*x1^4*x2*x3 + x1^4*x2 + x1^4*x3^5 + 3*x1^4*x3^4 + 2*x1^4*x3^3 - 10*x1^4*x3^2 + 3*x1^4*x3 - 11*x1^3*x2^3*x3^3 + 10*x1^3*x2^3*x3^2 - 3*x1^3*x2^3*x3 + 22*x1^3*x2^2*x3^3 - 16*x1^3*x2^2*x3^2 + 6*x1^3*x2^2*x3 + x1^3*x2*x3^5 + 7*x1^3*x2*x3^4 - 28*x1^3*x2*x3^3 + 11*x1^3*x2*x3^2 - 3*x1^3*x2*x3 - 3*x1^3*x3^5 - x1^3*x3^4 + 11*x1^3*x3^3 - 3*x1^3*x3^2 + 6*x1^2*x2^3*x3^4 - 3*x1^2*x2^3*x3^3 + 3*x1^2*x2^3*x3^2 - x1^2*x2^2*x3^5 - 17*x1^2*x2^2*x3^4 + 8*x1^2*x2^2*x3^3 - 8*x1^2*x2^2*x3^2 + x1^2*x2*x3^5 + 15*x1^2*x2*x3^4 - 2*x1^2*x2*x3^3 + 4*x1^2*x2*x3^2 + x1^2*x3^5 - 7*x1^2*x3^4 - x1*x2^3*x3^5 - 3*x1*x2^3*x3^3 + 4*x1*x2^2*x3^5 + 2*x1*x2^2*x3^4 + 6*x1*x2^2*x3^3 - 5*x1*x2*x3^5 - 4*x1*x2*x3^4 - 3*x1*x2*x3^3 + 2*x1*x3^5 + 2*x1*x3^4 + x2^3*x3^4 - x2^2*x3^5 - 2*x2^2*x3^4 + 2*x2*x3^5 + x2*x3^4 - x3^5, x1^4*x2 - x1^4 - x1^3*x2^2 - 3*x1^3*x2*x3 + 2*x1^3*x2 + x1^3*x3^2 + 2*x1^3*x3 - x1^3 + 3*x1^2*x2^2*x3 - 3*x1^2*x2*x3 - x1^2*x2 - 2*x1^2*x3^2 + 2*x1^2*x3 + x1^2 - x1*x2^2*x3^2 - x1*x2^2*x3 + 3*x1*x2*x3^2 + 2*x1*x2*x3 - x1*x3^2 - 2*x1*x3 - x2*x3^2 + x3^2, x1^3*x2^2 + x1^3*x2*x3 - 3*x1^3*x2 - x1^3*x3 + 2*x1^3 - 3*x1^2*x2^2*x3 - 2*x1^2*x2*x3^2 + 7*x1^2*x2*x3 + x1^2*x2 + x1^2*x3^3 - 3*x1^2*x3 - x1^2 + x1*x2^2*x3^2 + x1*x2^2*x3 + x1*x2*x3^3 - x1*x2*x3^2 - 4*x1*x2*x3 - 2*x1*x3^3 + 2*x1*x3^2 + 2*x1*x3 - x2*x3^3 + x2*x3^2 + x3^3 - x3^2, x1^3*x2*x3 - x1^3*x2 - x1^3*x3 + x1^3 + x1^2*x2^2 - 3*x1^2*x2*x3^2 + 2*x1^2*x2*x3 - x1^2*x2 + x1^2*x3^3 + x1^2*x3^2 - x1^2*x3 - 2*x1*x2^2*x3 + x1*x2*x3^3 + 2*x1*x2*x3^2 + x1*x2*x3 - 2*x1*x3^3 + x2^2*x3^2 - x2*x3^3 - x2*x3^2 + x3^3, x1^3*x2*x3 - x1^3*x2 - x1^3*x3 + x1^3 - x1^2*x2^2*x3 - 3*x1^2*x2*x3^2 + 4*x1^2*x2*x3 + x1^2*x2 + x1^2*x3^3 + x1^2*x3^2 - 2*x1^2*x3 - x1^2 + 3*x1*x2^2*x3^2 - x1*x2^2*x3 - 2*x1*x2*x3^2 - 2*x1*x2*x3 - x1*x3^3 + x1*x3^2 + 2*x1*x3 - x2^2*x3^3 + x2*x3^3 + x2*x3^2 - x3^2, x1^3*x2*x3 - x1^3*x2 - x1^3*x3 + x1^3 - x1^2*x2^2*x3 + 2*x1^2*x2^2 - 3*x1^2*x2*x3^2 + 4*x1^2*x2*x3 - 3*x1^2*x2 + x1^2*x3^3 + x1^2*x3^2 - 2*x1^2*x3 + x1^2 + 3*x1*x2^2*x3^2 - 5*x1*x2^2*x3 - 2*x1*x2*x3^2 + 6*x1*x2*x3 - x1*x3^3 + x1*x3^2 - 2*x1*x3 - x2^2*x3^3 + 2*x2^2*x3^2 + x2*x3^3 - 3*x2*x3^2 + x3^2]