The realization space is
  [1   1        x1^2   0   0   1        x1^2 + 1            x1^2          0            x1^2        x1^2 + 1]
  [0   1   x1^3 + x1   1   0   0        x1^2 + 1       x1^3 + x1   x1^2 + 1          x1 - 1       x1^3 + x1]
  [0   0           0   0   1   1   x1^2 - x1 + 1   x1^2 - x1 + 1         -1   x1^2 - x1 + 1   x1^2 - x1 + 1]
in the multivariate polynomial ring in 1 variable over ZZ
within the vanishing set of the ideal
Ideal with 1 generator
avoiding the zero loci of the polynomials
RingElem[x1^2 - x1 + 1, x1 + 1, x1, x1 - 1, x1^3 + x1^2 + 1, x1^3 - x1 + 1, x1^2 - x1 + 2, x1^2 + x1 + 1, x1^2 + 1]