```
3
- 2*x1 + x2 + 0.835634534*x1*(1-x1);
x1 - 2*x2 + x3 + 0.835634534*x2*(1-x2);
x2 - 2*x3      + 0.835634534*x3*(1-x3);

TITLE : 3-dimensional reaction-diffusion problem

ROOT COUNTS :

total degree : 8
mixed volume : 7

REFERENCES :

Communicated to me by Arieh Iserles, at the conference held
in Park City, Utah, July 1995.

The general formulation is as follows:

alpha > 0, x_0 = x_{n+1} = 0

f_k = x_{k-1} - 2*x_k + x_{k+1} + alpha*x_k*(1-x_k) = 0, k = 1,2,..,n.

It stems from a reaction diffusion problem.
For general dimension n, there are 2^n solutions, with the number of
real solutions increasing when the parameter alpha increases.
Though, there is only one real solution with all its components positive.
This parameter alpha is here the only real random constant in the system.

Note that the mixed volume does not count the trivial zero solution.

THE SOLUTIONS :

7 3
===========================================================
solution 1 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  9.90452246822526E-03   9.20668856728156E-01
x2 : -6.96695396506914E-01   1.08723496633471E+00
x3 : -1.40329531548205E+00  -9.20668856728156E-01
== err :  2.381E-15 = rco :  2.697E-01 = res :  3.140E-16 ==
solution 2 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.51954966926035E+00   1.43763339100755E+00
x2 : -1.56688585394284E+00  -1.97704910020807E+00
x3 : -1.51954966926035E+00   1.43763339100755E+00
== err :  6.957E-15 = rco :  2.882E-01 = res :  6.280E-16 ==
solution 3 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.40329531548205E+00  -9.20668856728156E-01
x2 : -6.96695396506914E-01   1.08723496633471E+00
x3 :  9.90452246822522E-03   9.20668856728156E-01
== err :  2.410E-15 = rco :  3.290E-01 = res :  4.965E-16 ==
solution 4 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.40329531548205E+00   9.20668856728156E-01
x2 : -6.96695396506914E-01  -1.08723496633471E+00
x3 :  9.90452246822524E-03  -9.20668856728156E-01
== err :  2.366E-15 = rco :  3.290E-01 = res :  3.140E-16 ==
solution 5 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  2.52317752493039E-01   9.48037483736659E-62
x2 :  3.46990121858033E-01   1.31266728517383E-61
x3 :  2.52317752493039E-01   9.48037483736659E-62
== err :  1.413E-15 = rco :  6.653E-02 = res :  1.110E-16 ==
solution 6 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.51954966926035E+00  -1.43763339100755E+00
x2 : -1.56688585394284E+00   1.97704910020807E+00
x3 : -1.51954966926035E+00  -1.43763339100755E+00
== err :  6.957E-15 = rco :  2.882E-01 = res :  6.280E-16 ==
solution 7 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  9.90452246822521E-03  -9.20668856728156E-01
x2 : -6.96695396506914E-01  -1.08723496633471E+00
x3 : -1.40329531548205E+00   9.20668856728156E-01
== err :  2.390E-15 = rco :  2.697E-01 = res :  3.514E-16 ==
```