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Method for solving a large system of under-determined equations?

I am trying to parametrize and find a family of solutions to some equations. (I am using the solve([eqns],vars) function.) Unfortunately, the equations are just complicated enough so that rather than parametrize, sage gives up and outputs the equations themselves.

Here is a (partial) particular example that I had in mind. My real equation is actually a bit more complicated. But this is a point where it went from solving to simply spitting out the original equations. Here is the output:

10 equations, and 12 unknowns. {s0 + s2 + s4: 1} {s5: 0} {s0 + s1 + s2 + s3 + s4 + s5: 2} {w0: s5} {w1: -s4} {w2: s3} {(s0*w0 + s1*w1)*(s0*w0 + s1*w1 + s2*w2): s2*w0 + s3*w1} {w3: -s2} {w4: s1} {w5: -s0}

Is there anything I can do? I've read the solve and x.solve pages, but I don't see a clear method I should try...

Method for solving a large system of under-determined equations?

I am trying to parametrize and find a family of solutions to some equations. (I am using the solve([eqns],vars) function.) Unfortunately, the equations are just complicated enough so that rather than parametrize, sage gives up and outputs the equations themselves.

Here is a (partial) particular example that I had in mind. My real equation is actually a bit more complicated. But this is a point where it went from solving to simply spitting out the original equations. Here is the output:

10 equations, and 12 unknowns. {s0 + s2 + s4: 1} {s5: 0} {s0 + s1 + s2 + s3 + s4 + s5: 2} {w0: s5} {w1: -s4} {w2: s3} {(s0*w0 + s1*w1)*(s0*w0 + s1*w1 + s2*w2): s2*w0 + s3*w1} {w3: -s2} {w4: s1} {w5: -s0}

Is there anything I can do? I've read the solve and x.solve pages, but I don't see a clear method I should try... try...

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Edit: I'm looking for the simplest representation of this system of equations. I would like Sage to parametrize a few of the variables and solve for the others. For instance, a linear system example would be the following:

eq1=a+b+c==2
eq2=b==1
solve([eq1,eq2],a,b,c)
The output would be:
a=1-r1, b=1, c=r1

Sage can do this using the solve function for small simple situations, even if they're non-linear equations. I want it to classify the family of all possible (real) solutions which satisfy all of the equations simultaneously. But it's stopping when I have a complicated system of equations.

Method for solving a large system of under-determined equations?

I am trying to parametrize and find a family of solutions to some equations. (I am using the solve([eqns],vars) function.) Unfortunately, the equations are just complicated enough so that rather than parametrize, sage gives up and outputs the equations themselves.

Here is a (partial) particular example that I had in mind. My real equation is actually a bit more complicated. But this is a point where it went from solving to simply spitting out the original equations. Here is the output:

10 equations, and 12 unknowns. {s0 + s2 + s4: 1} {s5: 0} {s0 + s1 + s2 + s3 + s4 + s5: 2} {w0: s5} {w1: -s4} {w2: s3} {(s0*w0 + s1*w1)*(s0*w0 + s1*w1 + s2*w2): s2*w0 + s3*w1} {w3: -s2} {w4: s1} {w5: -s0}

Is there anything I can do? I've read the solve and x.solve pages, but I don't see a clear method I should try...

===========

Edit: I'm looking for the simplest representation of this system of equations. I would like Sage to parametrize a few of the variables and solve for the others. For instance, a linear system example would be the following:

eq1=a+b+c==2
eq2=b==1
solve([eq1,eq2],a,b,c)
The output would be:
a=1-r1, b=1, c=r1

Sage can do this using the solve function for small simple situations, even if they're non-linear equations. I want it to classify the family of all possible (real) solutions which satisfy all of the equations simultaneously. But it's stopping when I have a complicated system of equations.

In essence I want it to "solve" the system of equations. Clearly this requires a few parameters, since there are more variables than equations... and since it's not a linear system I can't just ask a linear algebra student to do it. ;-)