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2010-12-22 19:09:42 +0100 commented answer Limitation of solve?

@Jason: Mathematica had trouble with the simplified version? That's surprising. I wonder why, since sage had no problem with it.

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2010-12-14 20:24:21 +0100 commented answer Limitation of solve?

This answer has already been extremely helpful, thanks. I guess at this point I'll have start using my brain as well as the computer to solve this problem :)

2010-12-13 13:06:16 +0100 commented question Limitation of solve?

Ah, much better. Thanks for the tip

2010-12-12 23:34:42 +0100 commented answer Limitation of solve?

Alright, I have updated and added the equations. Sorry, the simplified version was too simple so I put up the full version.

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2010-12-09 13:42:06 +0100 commented answer Limitation of solve?

What other information about my problem would be useful? And I do need algebraic solutions, not numeric approximates.

2010-12-07 23:21:47 +0100 asked a question Limitation of solve?

I am using (or trying to use) the solve function to solve a system of 10 nonlinear equations in 10 variables. However, solve simply outputs the 10 equations after some time computing. Is this running into a limit of the solve function, or is there some other way to economize my input to make it more solve-function-friendly? Would it help to know the algorithm which solve uses?

EDIT: At a suggestion from niles, I am putting up the equations that I am solving. I have a version that is simplified, but it is not a problem to solve. In the equations below it is legal to set any combination of the variables equal to zero, and the resulting solutions will be a subset of the solution set of the full collection of equations.

var('a b c d e f g h i j')

eq1 = a == a^2 + b*a + 2*a*c + 2*a*d + 2*a*e + 2*a*f + g*a + h*a + i*a + j*a + b^2 + d*b + g*b + b*c + d*c + g*c

eq2 = b == c*b + 2*b*e + h*b + b*d + c*d

eq3 = c == c^2 + 2*e*c + h*c + b*f + c*f

eq4 = d == f*b + d^2 + 2*e*d + f*d

eq5 = e == e^2

eq6 = f == f*c + i*c + 2*f*e + d*f + f^2

eq7 = g == i*b + j*b + 2*g*d + 2*h*d + i*d + j*d + 2*g*e + a*g + b*g + c*g + f*g + g^2 + 2*h*g + i*g + j*g + a*h + b*h + a*i

eq8 = h == 2*h*e + c*h + h^2 + b*i + c*i

eq9 = i == 2*i*e + f*h + i*h + d*i + f*i

eq10 = j == 2*j*e + g*f + h*f + j*f + 2*j*h + g*i + h*i + i^2 + j*i + a*j + b*j + c*j + d*j + f*j + g*j + i*j + j^2

For those who are curious, the simplified version is this set of equations with a, b, c, g, h, i, j == 0.

If there is anything more that anybody would like to know about these equations or where I am getting them from, just say so in a comment.

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2010-12-07 21:36:25 +0100 marked best answer Noncommuting variables

Perhaps FreeAlgebraQuotient will be useful? Here's the first part of the docstring:

sage: FreeAlgebraQuotient?
Type:           type
Base Class:     <type 'type'>
String Form:    <class 'sage.algebras.free_algebra_quotient.FreeAlgebraQuotient'>
Namespace:      Interactive
File:           /Applications/sage/local/lib/python2.6/site-packages/sage/algebras/free_algebra_quotient.py
Definition:     FreeAlgebraQuotient(self, x)
Docstring:

   Returns a quotient algebra defined via the action of a free algebra
   A on a (finitely generated) free module. The input for the quotient
   algebra is a list of monomials (in the underlying monoid for A)
   which form a free basis for the module of A, and a list of
   matrices, which give the action of the free generators of A on this
   monomial basis.

   EXAMPLES:

   Quaternion algebra defined in terms of three generators:

      sage: n = 3
      sage: A = FreeAlgebra(QQ,n,'i')
      sage: F = A.monoid()
      sage: i, j, k = F.gens()
      sage: mons = [ F(1), i, j, k ]
      sage: M = MatrixSpace(QQ,4)
      sage: mats = [M([0,1,0,0, -1,0,0,0, 0,0,0,-1, 0,0,1,0]),  M([0,0,1,0, 0,0,0,1, -1,0,0,0, 0,-1,0,0]),  M([0,0,0,1, 0,0,-1,0, 0,1,0,0, -1,0,0,0]) ]
      sage: H3.<i,j,k> = FreeAlgebraQuotient(A,mons,mats)
      sage: x = 1 + i + j + k
      sage: x
      1 + i + j + k
      sage: x**128
      -170141183460469231731687303715884105728 + 170141183460469231731687303715884105728*i + 170141183460469231731687303715884105728*j + 170141183460469231731687303715884105728*k
2010-12-07 21:34:22 +0100 commented answer Noncommuting variables

Thanks! This is just the thing

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2010-12-06 21:53:32 +0100 asked a question Noncommuting variables

I am extremely new to Sage, and even newer to this site, so I apologize if anything is not up to standards.

I am dealing with a multivariable polynomial ring over $\mathbb{Z}$ with noncommuting variables. Is there a way to implement this with Sage? The closest thing I have found is FreeAlgebra, where the variables are noncommutative, but I have not found any way to impose relations that I want.

As stated before, I am extremely new to all of this so don't assume that I know anything, and don't hesitate to give any and all suggestions.