var a;
var b;
var c;
var d;
var t;
var u;
var v;
var w;

subject to

cons1: a + b - 0.63254 = 0;
cons2: c + d + 1.34534 = 0;
cons3: t*a + u*b - v*c - w*d + 0.8365348 = 0;
cons4: v*a + w*b + t*c + u*d - 1.7345334 = 0;
cons5: a*t^2 - a*v^2 - 2*c*t*v + b*u^2 - b*w^2 - 2*d*u*w - 1.352352 = 0;
cons6: c*t^2 - c*v^2 + 2*a*t*v + d*u^2 - d*w^2 + 2*b*u*w + 0.843453 = 0;
cons7: a*t^3 - 3*a*t*v^2 + c*v^3 - 3*c*v*t^2 
  + b*u^3 - 3*b*u*w^2 + d*w^3 - 3*d*w*u^2 + 0.9563453 = 0;
cons8: c*t^3 - 3*c*t*v^2 - a*v^3 + 3*a*v*t^2 
  + d*u^3 - 3*d*u*w^2 - b*w^3 + 3*b*w*u^2 - 1.2342523 = 0;

solve;
display a, b, c, d, t, u, v, w;

# TITLE : heart-dipole problem

# ROOT COUNTS :

# total degree : 576
# 2-homogeneous Bezout number : 193
#   with partition : {a b c d }{t u v w }
# generalized Bezout number : 193
#   based on the set structure :
#      {a b }
#      {c d }
#      {a b c d }{t u v w }
#      {a b c d }{t u v w }
#      {a b c d }{t u v w }{t u v w }
#      {a b c d }{t u v w }{t u v w }
#      {a b c d }{t u v w }{t u v w }{t u v w }
#      {a b c d }{t u v w }{t u v w }{t u v w }
# mixed volume : 121

# REFERENCES :

# Nelsen, C.V. and Hodgkin, B.C.:
# `Determination of magnitudes, directions, and locations of two independent
#  dipoles in a circular conducting region from boundary potential measurements'
# IEEE Trans. Biomed. Engrg. Vol. BME-28, No. 12, pages 817-823, 1981.

# Morgan, A.P. and Sommese, A.J.:
# `Coefficient-Parameter Polynomial Continuation'
# Appl. Math. Comput. Vol. 29, No. 2, pages 123-160, 1989.
# Errata: Appl. Math. Comput. 51:207 (1992)

# Morgan, A.P. and Sommese, A. and Watson, L.T.:
# `Mathematical reduction of a heart dipole model'
# J. Comput. Appl. Math. Vol. 27, pages 407-410, 1989.

# SYMMETRY GROUP (FOR THE POLYTOPES ONLY!)

# 1
# (a b)(c d)
# (a b)(t u)(v w)
# (c d)(t u)(v w)
# (a c)(b d)(t v)(u w)
# (a d)(b c)
# (a d)(t w)(u v)
# (b c)(t w)(u v)

# a b c d t u v w

# 1 2 3 4 5 6 7 8
# 2 1 4 3 5 6 7 8
# 2 1 3 4 6 5 8 7
# 1 2 4 3 6 5 8 7
# 3 4 1 2 7 8 5 6
# 4 3 2 1 5 6 7 8
# 4 2 3 1 8 7 6 5
# 1 3 2 4 8 7 6 5

# NOTE :

# The deficiency of this system is due to the solutions of the homogeneous part.
# The Groebner bases for the homogeneous part contains the polynomials

#  { a + b, c+d , b*t - b*u - d*v + d*w, b*v - b*w + d*t - d*u,
#    d*t**2  - 2*d*t*u + d*u**2  + d*v**2  - 2*d*v*w + d*w**2 }

# which leads to four solution components at infinity :

#  1)  a=0, b=0, c=0, d=0, with t,u,v and w arbitrary complex numbers
#  2)  a= - d*i, b=d*i, c= - d, t= - i*v + i*w + u
#  3)  a=d*i, b= - d*i, c= - d, t=i*v - i*w + u
#  4)  a= - b, c= - d, t=u, v=w

# The consecutive contributions of the constant monomials to the mixed
# volume are as follows :

#    i   1    2    3    4    5    6    7    8
#   vi   0    0   65   65  104  104  121  121

# vi = mixed volume of the system with only the first i contant terms

# THE SOLUTIONS :

# 4 8
# ===========================================================
# solution 1 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  a :  3.16270000000000E-01  -9.40179958538160E-01
#  b :  3.16270000000000E-01   9.40179958538160E-01
#  c : -6.72670000000000E-01  -3.26802748753925E-01
#  d : -6.72670000000000E-01   3.26802748753925E-01
#  t :  1.05055232394915E-01   8.20732577685255E-02
#  u :  1.05055232394915E-01  -8.20732577685255E-02
#  v : -2.70836601845149E-01   1.06019057303938E+00
#  w : -2.70836601845149E-01  -1.06019057303938E+00
# == err :  2.975E-15 = rco :  4.003E-02 = res :  2.483E-16 ==
# solution 2 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  a : -1.05327487539249E-02  -2.40131204422346E-52
#  b :  6.43072748753925E-01   2.08809742975953E-52
#  c :  2.67509958538160E-01   1.04462108712268E-52
#  d : -1.61284995853816E+00  -1.04404871487976E-52
#  t :  1.16524580543429E+00  -7.51715074713430E-52
#  u : -9.55135340644462E-01   1.07014993275176E-52
#  v : -3.52909859613675E-01  -3.75857537356715E-52
#  w : -1.88763344076624E-01   8.35238971903811E-53
# == err :  2.694E-15 = rco :  4.064E-02 = res :  2.828E-16 ==
# solution 3 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  a :  6.43072748753925E-01  -3.18618382226490E-57
#  b : -1.05327487539250E-02   3.18618382226490E-57
#  c : -1.61284995853816E+00  -8.92131470234173E-57
#  d :  2.67509958538160E-01   8.92131470234173E-57
#  t : -9.55135340644462E-01   1.27447352890596E-57
#  u :  1.16524580543429E+00  -6.24492029163921E-56
#  v : -1.88763344076624E-01  -3.66411139560464E-57
#  w : -3.52909859613675E-01   3.82342058671789E-57
# == err :  2.465E-15 = rco :  3.546E-02 = res :  3.886E-16 ==
# solution 4 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  a :  3.16270000000000E-01   9.40179958538160E-01
#  b :  3.16270000000000E-01  -9.40179958538160E-01
#  c : -6.72670000000000E-01   3.26802748753925E-01
#  d : -6.72670000000000E-01  -3.26802748753925E-01
#  t :  1.05055232394915E-01  -8.20732577685255E-02
#  u :  1.05055232394915E-01   8.20732577685254E-02
#  v : -2.70836601845149E-01  -1.06019057303938E+00
#  w : -2.70836601845149E-01   1.06019057303938E+00
# == err :  3.067E-15 = rco :  4.003E-02 = res :  4.518E-16 ==