8
a + b - 0.63254;
c + d + 1.34534;
t*a + u*b - v*c - w*d + 0.8365348;
v*a + w*b + t*c + u*d - 1.7345334;
a*t**2 - a*v**2 - 2*c*t*v + b*u**2 - b*w**2 - 2*d*u*w - 1.352352;
c*t**2 - c*v**2 + 2*a*t*v + d*u**2 - d*w**2 + 2*b*u*w + 0.843453;
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;
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;
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 ==