9
c4 b2 b3 b4 a42 a43 a32 c3 c2   
/*  given by GBK heuristics. c2 c3 c4 b4 b3 b2 a32 a42 a43 */

2 b2 c2 + 2 b3 c3 + 2 b4 c4 -1 ;

3 b2 c2^2 + 3 b3 c3^2 + 3 b4 c4^2 - 1;

6 b3 a32 c2 + 6 b4 a42 c2 + 6 b4 a43 c3 - 1;

4 b2 c2^3 + 4 b3 c3^3 + 4 b4 c4^3 - 1;

8 b3 c3 a32 c2  + 8 b4 c4 a42 c2 + 8 b4 c4 a43 c3 - 1;

12 b3 a32 c2^2 + 12 b4 a42 c2^2 + 12 b4 c4 a43 c3^2 - 1;

24 b4 a43 a32 c2 -1;;

/* Hairer, Runge-Kutta 2, 05.11.83
** Found in "Some examples for Solving Systems of Algebraic Equations
** by Calculating Grobner Bases", by Boege, Gebauer and Kredel.
**   J. of Symbolic Computation (1986) 1,83-98.
**
** The other variables and equations listed in the article for this example
** do not need to be introduced in the computation of the GB because they 
** are given by linear equations with coefficients in the polynomial ring of 
** the variables listed above.
** 
** They are a41 given by c4 - a41 - a42 - a43, 
**	   b1  given by b1 + b2 + b3 + b4 - 1,
**	   a21 given by c2 - a21,
**	   a31 given by c3 - a31 - a32
*/








