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

subject to

cons1: x1*x2^2 + x1*x3^2 + x1*x4^2 - 1.1*x1 + 1 = 0;
cons2: x2*x1^2 + x2*x3^2 + x2*x4^2 - 1.1*x2 + 1 = 0;
cons3: x3*x1^2 + x3*x2^2 + x3*x4^2 - 1.1*x3 + 1 = 0;
cons4: x4*x1^2 + x4*x2^2 + x4*x3^2 - 1.1*x4 + 1 = 0;

solve;
display x1, x2, x3, x4;

# TITLE : A neural network modeled by an adaptive Lotka-Volterra system, n=4

# ROOT COUNTS :

# total degree : 81

# generalized Bezout bound : 73
#  with symmetric supporting set structure:
#   {x1 }{x2 x3 x4}{x2 x3 x4}
#   {x2 }{x1 x3 x4}{x1 x3 x4}
#   {x3 }{x1 x2 x4}{x1 x2 x4}
#   {x4 }{x1 x2 x3}{x1 x2 x3}

# mixed volume : 73

# REFERENCES :

# Karin Gatermann:
# "Symbolic solution of polynomial equation systems with symmetry",
# Proceedings of ISSAC-90, pp 112-119, ACM New York, 1990.

# V. W. Noonburg:
# "A neural network modeled by an adaptive Lotka-Volterra system",
# SIAM J. Appl. Math (1988).

# SYMMETRY :

# The coefficients have been chosen so that full permutation symmetry
# was obtained.  The parameter c = 1.1.

# ------------------------------------------------
# |    orbit information of list of solutions    |
# ------------------------------------------------
# |       TYPE        | NB <> | NB GEN | NB SOLS |
# ------------------------------------------------
# |  1 2 2 2          |   2   |    4   |    16   |
# |  1 1 2 3          |   3   |    3   |    36   |
# |  1 1 2 2          |   2   |    3   |    18   |
# |  1 1 1 1          |   1   |    3   |     3   |
# ------------------------------------------------
# | Total number of generating solutions :   13  |
# | Total number of solutions generated  :   73  |
# ------------------------------------------------

# THE GENERATING SOLUTIONS :

# 13 4
# ===========================================================
# solution 1 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  4.33372590962199E-01  -4.43586227384486E-01
#  x2 :  4.33372590962199E-01  -4.43586227384486E-01
#  x3 :  4.33372590962199E-01  -4.43586227384486E-01
#  x4 :  4.33372590962199E-01  -4.43586227384486E-01
# == err :  4.169E-16 = rco :  2.776E-01 = res :  3.608E-16 ==
# solution 2 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  1.01991909613079E+00   6.06272901205497E-17
#  x2 :  1.01991909613079E+00   6.06272901205497E-17
#  x3 : -5.09959548065397E-01  -8.48770395714556E-01
#  x4 : -5.09959548065397E-01   8.48770395714556E-01
# == err :  3.103E-15 = rco :  3.389E-01 = res :  2.269E-16 ==
# solution 3 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 : -2.14773924418920E-01  -6.26558193923668E-01
#  x2 : -2.14773924418920E-01  -6.26558193923668E-01
#  x3 :  1.06192282524925E+00  -3.99591785628118E-01
#  x4 :  1.06192282524925E+00  -3.99591785628118E-01
# == err :  3.638E-15 = rco :  2.138E-01 = res :  2.831E-16 ==
# solution 4 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 : -4.90152757143433E-01   2.95549546086774E-64
#  x2 : -1.20414504451724E+00  -1.37685732313517E-64
#  x3 : -4.90152757143432E-01   4.36796116304951E-64
#  x4 : -1.20414504451724E+00  -4.55787251796470E-64
# == err :  4.082E-15 = rco :  1.132E-01 = res :  3.331E-16 ==
# solution 5 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 : -6.72576917532508E-01  -3.25008188747211E-48
#  x2 : -9.28586522969462E-01   1.36845553156720E-48
#  x3 : -9.28586522969462E-01   5.98699295060652E-49
#  x4 : -9.28586522969462E-01   9.62195295633190E-49
# == err :  2.778E-16 = rco :  6.393E-02 = res :  2.220E-16 ==
# solution 6 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  1.35061583139308E+00   1.50901402336195E-01
#  x2 : -5.09959548065397E-01   4.79766841277292E-01
#  x3 : -5.09959548065397E-01   4.79766841277292E-01
#  x4 : -8.40656283327683E-01  -6.30668243613487E-01
# == err :  9.546E-16 = rco :  1.640E-01 = res :  2.220E-16 ==
# solution 7 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  1.35061583139308E+00  -1.50901402336195E-01
#  x2 : -8.40656283327683E-01   6.30668243613487E-01
#  x3 : -5.09959548065397E-01  -4.79766841277292E-01
#  x4 : -5.09959548065397E-01  -4.79766841277292E-01
# == err :  1.027E-15 = rco :  2.148E-01 = res :  4.965E-16 ==
# solution 8 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 : -9.17395351588499E-02  -1.00626933473987E+00
#  x2 :  6.75139752372438E-01  -2.43302218087769E-01
#  x3 :  6.75139752372438E-01  -2.43302218087769E-01
#  x4 :  6.75139752372438E-01  -2.43302218087769E-01
# == err :  1.124E-15 = rco :  2.482E-01 = res :  3.377E-16 ==
# solution 9 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 : -8.66745181924397E-01  -3.26265223399926E-53
#  x2 : -8.66745181924398E-01  -3.91518268079912E-54
#  x3 : -8.66745181924397E-01  -3.91518268079912E-53
#  x4 : -8.66745181924398E-01   7.96087145095820E-53
# == err :  5.989E-16 = rco :  6.026E-02 = res :  3.331E-16 ==
# solution 10 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  6.75139752372438E-01   2.43302218087769E-01
#  x2 :  6.75139752372438E-01   2.43302218087769E-01
#  x3 :  6.75139752372438E-01   2.43302218087769E-01
#  x4 : -9.17395351588496E-02   1.00626933473987E+00
# == err :  1.026E-15 = rco :  1.762E-01 = res :  5.495E-16 ==
# solution 11 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  1.06192282524925E+00   3.99591785628118E-01
#  x2 :  1.06192282524925E+00   3.99591785628118E-01
#  x3 : -2.14773924418920E-01   6.26558193923668E-01
#  x4 : -2.14773924418919E-01   6.26558193923668E-01
# == err :  3.602E-15 = rco :  2.457E-01 = res :  3.331E-16 ==
# solution 12 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  1.76514689694112E+00  -8.92131470234173E-57
#  x2 : -4.21692981775415E-01   5.22534146851444E-56
#  x3 : -4.21692981775415E-01  -2.35777602847603E-56
#  x4 : -4.21692981775415E-01  -1.68867742580040E-56
# == err :  3.707E-15 = rco :  4.190E-01 = res :  2.220E-16 ==
# solution 13 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  4.33372590962199E-01   4.43586227384486E-01
#  x2 :  4.33372590962199E-01   4.43586227384486E-01
#  x3 :  4.33372590962199E-01   4.43586227384486E-01
#  x4 :  4.33372590962199E-01   4.43586227384486E-01
# == err :  4.169E-16 = rco :  2.776E-01 = res :  3.608E-16 ==