This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.
There were 68 solutions found in 7.236 seconds (with a Bezout bound of 256).
Reference: "A product-decomposition theorem for bounding Bezout numbers" by A.P. Morgan, A.J. Sommese, and C.W. Wampler (pages 1308-1325).
See also: http://homepages.math.uic.edu/~jan/Demo/proddeco.html.
i1 : proddeco(RR_53)
2 2 2 2
o1 = {64.6329z z - 129.266z z z z + 64.6329z z - 55.6619z z z -
2 3 1 2 3 4 1 4 1 2 3
------------------------------------------------------------------------
2 2 2
12.201z z - 48.2115z z + 55.6619z z + 12.201z z z + 48.2115z z z -
2 3 2 3 1 4 1 2 4 1 3 4
------------------------------------------------------------------------
2
71.399z z z + 71.399z z + 50.7966z z + 9.14137z z + 62.3536z z +
2 3 4 1 4 1 3 2 3 1 4
------------------------------------------------------------------------
2 2 2 2
17.5403z z , 81.3704z z - 162.741z z z z + 81.3704z z - 55.3638z z z
2 4 2 3 1 2 3 4 1 4 1 2 3
------------------------------------------------------------------------
2 2 2
- 25.0326z z - 21.7819z z + 55.3638z z + 25.0326z z z +
2 3 2 3 1 4 1 2 4
------------------------------------------------------------------------
2
21.7819z z z - 55.0737z z z + 55.0737z z + 24.7967z z + 9.562z z +
1 3 4 2 3 4 1 4 1 3 2 3
------------------------------------------------------------------------
2 2 2 2
41.6882z z + 20.9301z z , 35.2817z z - 70.5634z z z z + 35.2817z z -
1 4 2 4 2 3 1 2 3 4 1 4
------------------------------------------------------------------------
2 2 2
42.2584z z z - 23.8847z z - 8.07978z z + 42.2584z z + 23.8847z z z
1 2 3 2 3 2 3 1 4 1 2 4
------------------------------------------------------------------------
2
+ 8.07978z z z - 23.1781z z z + 23.1781z z + 13.503z z + 10.3468z z
1 3 4 2 3 4 1 4 1 3 2 3
------------------------------------------------------------------------
2 2 2 2
+ 37.0338z z + 16.9465z z , 6.89936z z - 13.7987z z z z + 6.89936z z
1 4 2 4 2 3 1 2 3 4 1 4
------------------------------------------------------------------------
2 2 2
- 43.1035z z z - 56.0575z z - 2.37993z z + 43.1035z z +
1 2 3 2 3 2 3 1 4
------------------------------------------------------------------------
2
56.0575z z z + 2.37993z z z - 6.40051z z z + 6.40051z z +
1 2 4 1 3 4 2 3 4 1 4
------------------------------------------------------------------------
16.2879z z + 22.5597z z + 42.9769z z + 55.4137z z }
1 3 2 3 1 4 2 4
o1 : List
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The object proddeco is a method function.