This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-8250U CPU at 1.60GHz.
There were 40 solutions found in 0.546875 seconds (with a Bezout bound of 64).
Reference: Bellido A.M. "Construction of iteration functions for the simultaneous computation of the solutions of equations and algebraic systems." Numerical Algorithms, 6(3-4):315-351, 1992.
See also: http://www-sop.inria.fr/coprin/logiciels/ALIAS/Benches/node1.html#SECTION00012000000000000000
i1 : bellido(QQ)
2 2 2 2 2 2 2 2 2
o1 = {z + z + z - 12z - 68, z + z + z - 12z - 68, z + z + z - 24z
1 2 3 1 4 5 6 5 7 8 9 8
------------------------------------------------------------------------
- 12z + 100, z z + z z + z z - 6z - 6z - 52, z z + z z + z z -
9 1 4 2 5 3 6 1 5 1 7 2 8 3 9
------------------------------------------------------------------------
6z - 12z - 6z + 64, z z + z z + z z - 6z - 12z - 6z + 32, 2z +
1 8 9 4 7 5 8 6 9 5 8 9 2
------------------------------------------------------------------------
2z - z - z - 2z - z - z + 18, z + z + 2z + 2z + 2z - 2z + z
3 4 5 6 7 9 1 2 3 4 6 7 8
------------------------------------------------------------------------
- z - 38, z + z - 2z + z - z + 2z - 2z + 8}
9 1 3 4 5 6 7 8
o1 : List
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The object bellido is a method function.