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 16 solutions found in 0.531 seconds (with a Bezout bound of 128).
Reference: "Box-Bisection for solving second-degree systems and the problem of clustering', ACM Transactions on Mathematical Software" by A. Morgan and V. Shapiro. (p. 152-167).
See also: http://homepages.math.uic.edu/~jan/Demo/puma.html.
i1 : puma(RR_53)
2 2 2 2 2 2 2 2
o1 = {x + x - 1, x + x - 1, x + x - 1, x + x - 1, .004731x x -
1 2 3 4 5 6 7 8 1 3
------------------------------------------------------------------------
.3578x x - .1238x - .001637x - .9338x + x - .3571, .2238x x +
2 3 1 2 4 7 1 3
------------------------------------------------------------------------
.7623x x + .2638x - .07745x - .6734x - .6022, x x + .3578x +
2 3 1 2 4 6 8 1
------------------------------------------------------------------------
.004731x , - .7623x + .2238x + .3461}
2 1 2
o1 : List
|
The object puma is a method function.