N=1,   # of fermion fields: 1,   # of boson fields: 0
weight(t)=10,   weight(s)=12,   fermion weights={2},   boson weights={}


Problem | Unknowns | Inequalities | Equations | Solution 1 | Computing time | Back to overview

Problem

Find equations

                          2                           2
f  := Df  *Df*f*p1 + (Df ) *f*p2 + Df *Df*f *p3 + (Df) *f  *p4 + f  *p6
 t      2x              x            x     x             2x       5x

       + f  *f *f*p5
          2x  x

with symmetries
                                                          2
f  := Df  *Df*f*q3 + Df  *Df *f*q2 + Df  *Df*f *q4 + (Df ) *f *q5
 s      3x             2x   x          2x     x         x    x

                             4            2
       + Df *Df*f  *q6 + (Df) *f*q1 + (Df) *f  *q7 + f  *q9 + f  *f *f*q8
           x     2x                          3x       6x       3x  x

Unknowns

All solutions for the following 15 unknowns have to be determined:
p1,p2,p3,p4,p5,p6,q1,q2,q3,q4,q5,q6,q7,q8,q9

Inequalities

Each of the following lists represents one inequality which states that not all unknowns in this list may vanish. These inequalities filter out solutions which are trivial for the application.
{q8,q7,q6,q5,q4,q3,q2,q1,p5,p4,p3,p2,p1}
{q9,q8,q7,q6,q5,q4,q3,q2,q1}
{p6,p5,p4,p3,p2,p1}

Equations

All comma separated 55 expressions involving 409 terms have to vanish.
p4*q1,
p1*q1,
3*q1*(p3 + 4/3*p4),
p4*q7 - 10*p6*q1,
16*(p5*q9 - 15/16*p6*q8),
p5*q9 - p6*q8,
9*(p5*q9 - 10/9*p6*q8),
6*(p5*q9 - 5/6*p6*q8),
6*(p4*q9 - 5/6*p6*q7),
6*(p1*q9 - 5/6*p6*q3),
10*q1*(p1 + 3/10*p2 + 3/5*p4),
2*q1*(p1 - 2*p3 + 4*p4 - 1/2*p5),
p2*q5 + p3*q5 - 60*p6*q1,
2*(p2*q6 + 3/2*p3*q6 + p4*q6 - 120*p6*q1),
6*(p3*q9 + 5*p4*q9 - 5/6*p6*q6 - 10/3*p6*q7),
3*(p1*q9 + 4/3*p2*q9 - 2/3*p6*q2 - 1/3*p6*q3),
26*(p1*q9 + 15/13*p2*q9 - 15/26*p6*q2 - 5/13*p6*q3),
15*(p1*q9 + 2/5*p3*q9 - 2/3*p6*q3 - 1/3*p6*q4),
15*(p1*q9 + 4/5*p2*q9 - 1/3*p6*q2 - 2/3*p6*q3),
2*(p2*q7 + 3/2*p3*q7 + p4*q6 + 2*p4*q7 - 60*p6*q1),
2*(p1*q7 + 3/2*p3*q7 - p4*q6 + 3*p4*q7 - 20*p6*q1),
p1*q4 + 3*p1*q7 - p3*q3 - 2*p4*q3 - 20*p6*q1,
p1*q6 + 3*p1*q7 + 3*p3*q7 - 2*p4*q3 - 2*p4*q4 - 40*p6*q1,
2*(p1*q3 - 1/2*p1*q4 - 1/2*p1*q7 + 1/2*p3*q3 + 2*p4*q3 - 30*p6*q1),
6*(p1*q9 + 5/2*p3*q9 + 20/3*p4*q9 - 5/6*p6*q4 - 5/3*p6*q6 - 10/3*p6*q7),
20*(p1*q9 + 3/4*p3*q9 + 3/5*p4*q9 - 1/2*p6*q3 - 1/2*p6*q4 - 1/4*p6*q6),
30*(p1*q9 + 2/5*p2*q9 + 1/5*p3*q9 - 1/6*p6*q2 - 2/3*p6*q3 - 1/6*p6*q4),
p1*q3 - 2*p1*q4 + 2*p1*q7 - p1*q8 + 2*p3*q3 - 2*p4*q3 + p5*q3,
3*(p1*q9 + 4/3*p3*q9 + 2*p4*q9 - 1/3*p6*q3 - 2/3*p6*q4 - 2/3*p6*q6 - 2/3*p6*q7),
18*(p1*q9 + 4/3*p2*q9 + 7/18*p3*q9 - 5/9*p6*q2 - 5/18*p6*q3 - 1/6*p6*q4 - 2/9*p6
*q5),
66*(p1*q9 + 10/11*p2*q9 + 7/22*p3*q9 - 25/66*p6*q2 - 5/11*p6*q3 - 5/33*p6*q4 - 5
/33*p6*q5),
p1*q2 + 2*p2*q2 - p2*q4 - 2*p2*q5 - p2*q6 + 3*p3*q2 + p4*q2 - 120*p6*q1,
6*(p1*q9 + 2*p2*q9 + 15/2*p3*q9 + 20*p4*q9 - 5/6*p6*q4 - 5/3*p6*q5 - 5*p6*q6 - 
10*p6*q7),
3*(p1*q9 + 2*p2*q9 + 4*p3*q9 + 6*p4*q9 - 1/3*p6*q2 - 2/3*p6*q4 - 4/3*p6*q5 - 2*
p6*q6 - 2*p6*q7),
6*(p1*q9 + 3/2*p2*q9 + p3*q9 + 1/3*p4*q9 - 1/2*p6*q2 - 1/6*p6*q3 - 1/3*p6*q4 - 1
/2*p6*q5 - 1/6*p6*q6),
21*(p1*q9 + 8/7*p2*q9 + 6/7*p3*q9 + 2/7*p4*q9 - 8/21*p6*q2 - 2/7*p6*q3 - 1/3*p6*
q4 - 8/21*p6*q5 - 1/7*p6*q6),
60*(p1*q9 + 1/2*p2*q9 + 3/5*p3*q9 + 1/5*p4*q9 - 1/6*p6*q2 - 1/2*p6*q3 - 1/3*p6*
q4 - 1/6*p6*q5 - 1/12*p6*q6),
p1*q4 - p1*q6 - p3*q6 + 4*p3*q7 - p3*q8 + 2*p4*q6 - 8*p4*q7 + p5*q4 - p5*q6 + 2*
p5*q7,
2*(p1*q3 - p1*q4 - 1/2*p1*q6 + 2*p1*q7 - p1*q8 + p3*q3 - p4*q3 - p4*q8 + p5*q3 +
 3/2*p5*q7),
6*(p1*q9 + p2*q9 + 8/3*p3*q9 + 4*p4*q9 - 1/6*p6*q2 - 1/6*p6*q3 - 2/3*p6*q4 - 2/3
*p6*q5 - 4/3*p6*q6 - 4/3*p6*q7),
8*(p1*q9 + 3/2*p2*q9 + 15/8*p3*q9 + 3/2*p4*q9 - 3/8*p6*q2 - 1/8*p6*q3 - 1/2*p6*
q4 - 3/4*p6*q5 - 5/8*p6*q6 - 1/4*p6*q7),
12*(p1*q9 + 2/3*p2*q9 + 5/4*p3*q9 + p4*q9 - 1/6*p6*q2 - 1/3*p6*q3 - 1/2*p6*q4 - 
1/3*p6*q5 - 5/12*p6*q6 - 1/6*p6*q7),
3*(p1*q6 + p1*q7 + 2/3*p2*q6 + 2*p2*q7 + p3*q6 + 3*p3*q7 - 2/3*p4*q2 - 2/3*p4*q4
 - 4/3*p4*q5 + 2*p4*q7 - 120*p6*q1),
6*(p1*q3 - 1/2*p1*q4 - p1*q5 - 1/2*p1*q6 + 1/3*p2*q2 + 2*p2*q3 + 1/2*p3*q2 + 2*
p3*q3 + 2/3*p4*q2 + p4*q3 - 60*p6*q1),
2*(p1*q2 - 3*p1*q3 + p1*q4 + 3/2*p1*q6 - 1/2*p1*q7 - 3*p2*q3 - 3*p2*q7 - 3*p3*q3
 + p4*q2 - 4*p4*q3 + 60*p6*q1),
2*(p1*q4 + 2*p1*q5 + p1*q6 + 3*p2*q4 + 2*p2*q5 + 2*p2*q6 - p3*q2 + 2*p3*q4 + 1/2
*p3*q6 + p4*q4 + 2*p4*q5 - 360*p6*q1),
3*(p1*q4 + p1*q7 + 2/3*p2*q4 + 2*p2*q7 - 1/3*p3*q2 + 2/3*p3*q4 - 1/3*p3*q6 + p3*
q7 - 2/3*p4*q2 + 2/3*p4*q4 - 2/3*p4*q5 - 60*p6*q1),
2*(p1*q2 - p1*q3 + p1*q4 + p1*q5 - p1*q6 - p1*q8 - 3*p2*q3 + p2*q4 - 1/2*p3*q2 -
 3/2*p3*q3 - 2*p3*q8 + 3*p4*q3 + 2*p5*q4),
4*(p1*q4 + 1/2*p1*q5 + 1/2*p1*q6 + 3/2*p1*q7 + 1/2*p2*q4 + 3/2*p2*q7 - 1/4*p3*q2
 - 3/4*p3*q3 + 1/4*p3*q7 - 1/2*p4*q2 - 1/2*p4*q3 + 1/2*p4*q4 - 60*p6*q1),
2*(p1*q2 + 2*p1*q3 - p1*q4 - p1*q5 - p1*q7 + 2*p2*q3 - p2*q4 - 2*p2*q7 + 1/2*p3*
q2 + 7/2*p3*q3 + 2*p4*q2 + 3*p4*q3 - 120*p6*q1),
2*(p1*q2 - p1*q5 + p2*q2 - p2*q4 - 4*p2*q5 + p2*q6 + 2*p2*q7 + p2*q8 + 3/2*p3*q2
 + 3/2*p3*q8 - 2*p4*q2 + 3*p4*q8 + p5*q5 - p5*q6 - p5*q7),
3*(p1*q3 - 2/3*p1*q4 - 2/3*p1*q5 - 1/3*p1*q6 + 4/3*p1*q7 - 1/3*p1*q8 + 4/3*p2*q3
 - 2/3*p2*q4 - 2/3*p2*q6 + 8/3*p2*q7 - 2/3*p2*q8 + 1/3*p3*q2 + p3*q3 - 2*p4*q3 +
 1/3*p5*q2 + 1/3*p5*q3 + p5*q6),
p1*q2 - 4*p1*q5 + 2*p1*q6 + 3*p1*q8 + 4*p2*q2 - 2*p2*q4 - 8*p2*q5 + 2*p2*q6 + 6*
p2*q8 + 3*p3*q2 + 9*p3*q8 - 4*p4*q2 + 6*p4*q8 - 2*p5*q2 - 2*p5*q4 - 6*p5*q5 - 2*
p5*q6,
4*(p1*q3 - p1*q4 - p1*q5 + 1/4*p1*q6 + p1*q7 + p2*q3 - p2*q4 + p2*q7 - 1/2*p2*q8
 + 1/2*p3*q2 + 5/4*p3*q3 - 1/4*p3*q8 - 1/2*p4*q2 - 3/2*p4*q3 + 3/2*p4*q8 + 1/4*
p5*q2 + 1/4*p5*q4 - p5*q7),
3*(p1*q2 - 5/3*p1*q3 + 5/3*p1*q4 + 4/3*p1*q5 - 2/3*p1*q6 - 2*p1*q7 - 2/3*p1*q8 -
 4*p2*q3 + 4/3*p2*q4 - p3*q2 - p3*q3 - p3*q8 + 2/3*p4*q2 + 2*p4*q3 - 2*p4*q8 - 1
/3*p5*q2 + 1/3*p5*q3 - 1/3*p5*q4 + 2/3*p5*q6)

Computing time

On a Pentium 4 PC with 1.7GHz running REDUCE 3.7 with 120 MB RAM under Linux it took 59 sec.