Souleiman omar hoche wrote:
>
> Hi i want to ask if it is possible to solve the sytem of equations below
> with fricas:
>
> eq1:= po^2+2*p1*p4+2*p2*p3+qo^2+q1^2+q2^2+q3^2+q4^2-po=0;
>
> eq2:=2*po*p1 +2*p2*p4+p3^2+qo*q4+q1*qo+q2*q1+q3*q2+q4*q3-p1=0;
>
> eq3:= 2*po*p2+p1^2+2*p3*p4+qo*q3+q1*q4+q2*qo+q3*q1+q4*q2-p2=0;
>
> eq4:=2*po*p3+2*p1*p2+p4^2+qo*q2+q1*q3+q2*q4+q3*q0+q4*q1-p3=0;
^^
Do you mean q0 or maybe qo here?
>
> eq5:=2*po*p4+2*p1*p3+p2^2+qo*q1+q1*q2+q2*q3+q3*q4+q4*qo-p4=0;
>
> eq6:=2*po*qo+p1*q4+p2*q3+p3*q2+p4*q1+q1*p1+q2*p2+q3*p3+q4*p4-qo=0;
>
> eq7:=2*po*q1+p1*qo+p2*q4+p3*q3+p4*q2+qo*p4+q2*p1+q3*p2+q4*p3-q1=0;
>
> eq8:=2*po*q2+p1*q1+p2*qo+p3*q4+p4*q3+qo*p3+q1*p4+q3*p1+q4*p2-q2=0;
>
> eq9:=2*po*q3+p1*q2+p2*q1+p3*qo+p4*q4+qo*p2+q1*p3+q2*p4+q4*p1-q3=0;
>
> eq10:=2*po*q4+p1*q3+p2*q2+p3*q1+p4*qo+qo*p1+q1*p2+q2*p3+q3*p4-q4=0;
>
> I tried to do it with the function *radicalSolve* and *solve*, and i got
> the message: "*Error detected within library code: system does not have a
> finite number of solutions*".
>
> I want to precise that i first tried to do it with *matlab *and *maple *and
> i got two different possible solution one with matlab (po=1/10; p1=1/10;
> p2=1/10; p3=1/10; p4=1/10; qo=1/10; q1=1/10; q2=1/10; q3=1/10;
> q4=1/10) and the other one with maple
>
> (p1 = 0.03344940640, p2 = -0.06720617766, p3 = 0.06720617766, p4 =
> -0.03344940640, po = 0.5000000000, q1 = -0.04300931701, q2 = -0.3199584031,
> q3 = -0.2454933756, q4 = 0.2674194114, qo = -0.1589583157).
>
> Also another possible solution (po=1/10; p1=1/10; p2=1/10; p3=1/10;
> p4=1/10; qo=-1/10; q1=-1/10; q2=-1/10; q3=-1/10; q4=-1/10).
>
>
> I need all the solution (if it is possible). Do you think that fricas can
> help for this? If yes, wich command or function do you advice me?
With q0 in eq4 values above fail to solve the system, so I assume
you want qo. Using:
leq := [eq1, eq2, eq3, eq4, eq5, eq6, eq7, eq8, eq9, eq10];
lp := [lhs(eq) for eq in leq]
vars := [p1, p2, p3, p4, po, q1, q2, q3, q4, qo]
hd := HDMP(vars, Integer)
lp1 := [eq::hd for eq in lp]
gbf := groebnerFactorize(lp1)
I get 119 simpler systems. Solutions to your original system
are union of solutions of those 119 systems. Some systems
directy give solutions, for example:
[10p1 + 1, 10p2 + 1, 10p3 + 1, 10p4 + 1, 10po - 9, 10q1 + 1, 10q2 + 1,
10q3 + 1, 10q4 + 1, 10qo + 1]
,
[10p1 - 1, 10p2 - 1, 10p3 - 1, 10p4 - 1, 10po - 1, 10q1 + 1, 10q2 + 1,
10q3 + 1, 10q4 + 1, 10qo + 1]
,
[10p1 + 1, 10p2 + 1, 10p3 + 1, 10p4 + 1, 10po - 9, 10q1 - 1, 10q2 - 1,
10q3 - 1, 10q4 - 1, 10qo - 1]
,
[10p1 - 1, 10p2 - 1, 10p3 - 1, 10p4 - 1, 10po - 1, 10q1 - 1, 10q2 - 1,
10q3 - 1, 10q4 - 1, 10qo - 1]
,
[10p1 + 1, 10p2 + 1, 10p3 + 1, 10p4 + 1, 5po - 2, 5q1 + 2, 10q2 - 1,
10q3 - 1, 10q4 - 1, 10qo - 1]
,
[10p1 - 1, 10p2 - 1, 10p3 - 1, 10p4 - 1, 5po - 3, 5q1 + 2, 10q2 - 1,
10q3 - 1, 10q4 - 1, 10qo - 1]
,
[p1,p2,p3,p4,po,q1,q2,q3,q4,qo], [p1,p2,p3,p4,po - 1,q1,q2,q3,q4,qo]
give 8 rational solutions. Other systems, like:
2 2
[5p4 - 4q2 + 2q2,p1 + p4,2p2 - p4,2p3 + p4,2po - 1,2q1 + 2q2 - 1,q3,q4,qo]
almost directly give 1 dimensional family of solutions (the first polynomial
give a curve in (p4, q2) plane and other equations are linear and have
unique solution for any (p4, q2)). Other system require more work
to solve, but typically some variables are determined so should
be easier than your original system. Most seem to have infinite
number of solutions and FriCAS is of limited help with them.
OTOH, I do not know in which form you would like to see solutions:
in some cases solution set has nice parametrization, but this is
not always the case. BTW, there is system consisting of only 7
equations, so you will get at least 3 dimensional solution set.
--
Waldek Hebisch
[email protected]
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