var x1;
var x2;
var x3;
var x4;

subject to

cons1: x1*x2-x1*x3-x4+ 1 = 0;
cons2: x2*x3-x2*x4-x1+ 1 = 0;
cons3: -x1*x3+x3*x4-x2+ 1 = 0;
cons4: x1*x4-x2*x4-x3+ 1 = 0;

solve;
display x1, x2, x3, x4;

# TITLE : equilibrium points of a 4-dimensional Lorentz attractor

# ROOT COUNTS :

# total degree : 16
# 2-homogeneous Bezout number : 14
#   with partition : {x1 x2 }{x3 x4 }
# generalized Bezout number : 12
#   based on the set structure :
#      {x1 x4 }{x2 x3 }
#      {x1 x2 }{x3 x4 }
#      {x1 x2 x4 }{x3 }
#      {x1 x2 x3 }{x4 }
# mixed volume : 12

# REFERENCES :

# Tien-Yien Li : "Solving polynomial systems",
# The Mathematical Intelligencer 9(3):33-39, 1987.

# THE SOLUTIONS :

# 11 4
# ===========================================================
# solution 1 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  3.09016994374947E-01   9.51056516295154E-01
#  x2 : -8.09016994374948E-01   5.87785252292473E-01
#  x3 :  3.09016994374947E-01  -9.51056516295154E-01
#  x4 : -8.09016994374948E-01  -5.87785252292473E-01
# == err :  1.021E-15 = rco :  3.839E-02 = res :  2.483E-16 ==
# solution 2 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  1.00000000000000E+00   0.00000000000000E+00
#  x2 :  1.00000000000000E+00  -3.82342058671789E-57
#  x3 :  1.00000000000000E+00  -1.91171029335894E-57
#  x4 :  1.00000000000000E+00  -1.27447352890596E-57
# == err :  1.291E-41 = rco :  3.375E-01 = res :  5.735E-57 ==
# solution 3 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 : -1.00000000000000E+00   1.00000000000000E+00
#  x2 : -5.12631296267255E-17  -1.00000000000000E+00
#  x3 : -1.00000000000000E+00  -1.00000000000000E+00
#  x4 : -4.55035905491173E-17   1.00000000000000E+00
# == err :  0.000E+00 = rco :  4.454E-02 = res :  0.000E+00 ==
# solution 4 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 : -1.81709374138875E-17  -1.00000000000000E+00
#  x2 : -1.00000000000000E+00  -1.00000000000000E+00
#  x3 :  2.38907385022240E-17   1.00000000000000E+00
#  x4 : -1.00000000000000E+00   1.00000000000000E+00
# == err :  3.349E-17 = rco :  3.811E-02 = res :  4.206E-17 ==
# solution 5 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  3.09016994374947E-01  -9.51056516295154E-01
#  x2 : -8.09016994374948E-01  -5.87785252292473E-01
#  x3 :  3.09016994374947E-01   9.51056516295154E-01
#  x4 : -8.09016994374948E-01   5.87785252292473E-01
# == err :  6.687E-16 = rco :  3.839E-02 = res :  2.483E-16 ==
# solution 6 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  1.71463975042395E-17   1.00000000000000E+00
#  x2 : -1.00000000000000E+00   1.00000000000000E+00
#  x3 : -3.44326771178718E-17  -1.00000000000000E+00
#  x4 : -1.00000000000000E+00  -1.00000000000000E+00
# == err :  2.177E-16 = rco :  3.811E-02 = res :  4.578E-16 ==
# solution 7 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  0.00000000000000E+00   0.00000000000000E+00
#  x2 :  1.00000000000000E+00   0.00000000000000E+00
#  x3 : -6.48149604096329E-18   0.00000000000000E+00
#  x4 :  1.00000000000000E+00   0.00000000000000E+00
# == err :  1.148E-41 = rco :  3.333E-01 = res :  0.000E+00 ==
# solution 8 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 : -8.09016994374948E-01  -5.87785252292473E-01
#  x2 :  3.09016994374948E-01   9.51056516295154E-01
#  x3 : -8.09016994374947E-01   5.87785252292473E-01
#  x4 :  3.09016994374947E-01  -9.51056516295153E-01
# == err :  9.148E-16 = rco :  3.574E-02 = res :  2.220E-16 ==
# solution 9 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 : -8.09016994374948E-01   5.87785252292473E-01
#  x2 :  3.09016994374947E-01  -9.51056516295154E-01
#  x3 : -8.09016994374947E-01  -5.87785252292473E-01
#  x4 :  3.09016994374947E-01   9.51056516295153E-01
# == err :  6.454E-16 = rco :  3.574E-02 = res :  2.483E-16 ==
# solution 10 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  1.00000000000000E+00   6.96997109758186E-61
#  x2 :  5.84394251940559E-17   6.42673050142134E-61
#  x3 :  1.00000000000000E+00  -1.24460305557223E-60
#  x4 : -9.69173609317895E-18   6.22301527786114E-61
# == err :  9.692E-18 = rco :  2.000E-01 = res :  1.110E-16 ==
# solution 11 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 : -1.00000000000000E+00  -1.00000000000000E+00
#  x2 : -2.25656724857668E-17   1.00000000000000E+00
#  x3 : -1.00000000000000E+00   1.00000000000000E+00
#  x4 : -5.35036568346616E-17  -1.00000000000000E+00
# == err :  0.000E+00 = rco :  4.454E-02 = res :  0.000E+00 ==
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