Interfaces#

A central facet of Sage is that it supports computation with objects in many different computer algebra systems “under one roof” using a common interface and clean programming language.

The console and interact methods of an interface do very different things. For example, using GAP as an example:

gap.console(): This opens the GAP console - it transfers control to GAP. Here Sage is serving as nothing more than a convenient program launcher, similar to the Linux bash shell.
gap.interact(): This is a convenient way to interact with a running GAP instance that may be “full of” Sage objects. You can import Sage objects into this GAP session (even from the interactive interface), etc.

GAP#

Sage comes with GAP for computational discrete mathematics, especially group theory.

Here’s an example of GAP’s IdGroup function.

Sage

sage: G = gap('Group((1,2,3)(4,5), (3,4))')
sage: G
Group( [ (1,2,3)(4,5), (3,4) ] )
sage: G.Center()
Group( () )
sage: G.IdGroup()
[ 120, 34 ]
sage: G.Order()
120

Python

>>> from sage.all import *
>>> G = gap('Group((1,2,3)(4,5), (3,4))')
>>> G
Group( [ (1,2,3)(4,5), (3,4) ] )
>>> G.Center()
Group( () )
>>> G.IdGroup()
[ 120, 34 ]
>>> G.Order()
120

We can do the same computation in Sage without explicitly invoking the GAP interface as follows:

Sage

sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: G.center()
Subgroup generated by [()] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)])
sage: G.group_id()
[120, 34]
sage: n = G.order(); n
120

Python

>>> from sage.all import *
>>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))],[(Integer(3),Integer(4))]])
>>> G.center()
Subgroup generated by [()] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)])
>>> G.group_id()
[120, 34]
>>> n = G.order(); n
120

For some GAP functionality, you should install an optional Sage packages. This can be done with the command:

sage -i gap_packages

Singular#

Singular provides a massive and mature library for Gröbner bases, multivariate polynomial gcds, bases of Riemann-Roch spaces of a plane curve, and factorizations, among other things. We illustrate multivariate polynomial factorization using the Sage interface to Singular (do not type the ....:):

Sage

sage: R1 = singular.ring(0, '(x,y)', 'dp')
sage: R1
polynomial ring, over a field, global ordering
//   coefficients: QQ
//   number of vars : 2
//        block   1 : ordering dp
//                  : names    x y
//        block   2 : ordering C
sage: f = singular('9*y^8 - 9*x^2*y^7 - 18*x^3*y^6 - 18*x^5*y^6 +'
....:     '9*x^6*y^4 + 18*x^7*y^5 + 36*x^8*y^4 + 9*x^10*y^4 - 18*x^11*y^2 -'
....:     '9*x^12*y^3 - 18*x^13*y^2 + 9*x^16')

Python

>>> from sage.all import *
>>> R1 = singular.ring(Integer(0), '(x,y)', 'dp')
>>> R1
polynomial ring, over a field, global ordering
//   coefficients: QQ
//   number of vars : 2
//        block   1 : ordering dp
//                  : names    x y
//        block   2 : ordering C
>>> f = singular('9*y^8 - 9*x^2*y^7 - 18*x^3*y^6 - 18*x^5*y^6 +'
...     '9*x^6*y^4 + 18*x^7*y^5 + 36*x^8*y^4 + 9*x^10*y^4 - 18*x^11*y^2 -'
...     '9*x^12*y^3 - 18*x^13*y^2 + 9*x^16')

Now that we have defined \(f\), we print it and factor.

Sage

sage: f
9*x^16-18*x^13*y^2-9*x^12*y^3+9*x^10*y^4-18*x^11*y^2+36*x^8*y^4+18*x^7*y^5-18*x^5*y^6+9*x^6*y^4-18*x^3*y^6-9*x^2*y^7+9*y^8
sage: f.parent()
Singular
sage: F = f.factorize(); F
[1]:
   _[1]=9
   _[2]=x^6-2*x^3*y^2-x^2*y^3+y^4
   _[3]=-x^5+y^2
[2]:
   1,1,2
sage: F[1][2]
x^6-2*x^3*y^2-x^2*y^3+y^4

Python

>>> from sage.all import *
>>> f
9*x^16-18*x^13*y^2-9*x^12*y^3+9*x^10*y^4-18*x^11*y^2+36*x^8*y^4+18*x^7*y^5-18*x^5*y^6+9*x^6*y^4-18*x^3*y^6-9*x^2*y^7+9*y^8
>>> f.parent()
Singular
>>> F = f.factorize(); F
[1]:
   _[1]=9
   _[2]=x^6-2*x^3*y^2-x^2*y^3+y^4
   _[3]=-x^5+y^2
[2]:
   1,1,2
>>> F[Integer(1)][Integer(2)]
x^6-2*x^3*y^2-x^2*y^3+y^4

As with the GAP example in GAP, we can compute the above factorization without explicitly using the Singular interface (however, behind the scenes Sage uses the Singular interface for the actual computation). Do not type the ....::

Sage

sage: x, y = QQ['x, y'].gens()
sage: f = (9*y^8 - 9*x^2*y^7 - 18*x^3*y^6 - 18*x^5*y^6 + 9*x^6*y^4
....:     + 18*x^7*y^5 + 36*x^8*y^4 + 9*x^10*y^4 - 18*x^11*y^2 - 9*x^12*y^3
....:     - 18*x^13*y^2 + 9*x^16)
sage: factor(f)
(9) * (-x^5 + y^2)^2 * (x^6 - 2*x^3*y^2 - x^2*y^3 + y^4)

Python

>>> from sage.all import *
>>> x, y = QQ['x, y'].gens()
>>> f = (Integer(9)*y**Integer(8) - Integer(9)*x**Integer(2)*y**Integer(7) - Integer(18)*x**Integer(3)*y**Integer(6) - Integer(18)*x**Integer(5)*y**Integer(6) + Integer(9)*x**Integer(6)*y**Integer(4)
...     + Integer(18)*x**Integer(7)*y**Integer(5) + Integer(36)*x**Integer(8)*y**Integer(4) + Integer(9)*x**Integer(10)*y**Integer(4) - Integer(18)*x**Integer(11)*y**Integer(2) - Integer(9)*x**Integer(12)*y**Integer(3)
...     - Integer(18)*x**Integer(13)*y**Integer(2) + Integer(9)*x**Integer(16))
>>> factor(f)
(9) * (-x^5 + y^2)^2 * (x^6 - 2*x^3*y^2 - x^2*y^3 + y^4)

Interfaces#

GP/PARI#

GAP#

Singular#

Maxima#