Interface to Singular

AUTHORS:

  • David Joyner and William Stein (2005): first version
  • Martin Albrecht (2006-03-05): code so singular.[tab] and x = singular(...), x.[tab] includes all singular commands.
  • Martin Albrecht (2006-03-06): This patch adds the equality symbol to singular. Also fix a problem in which ” ” as prompt means comparison will break all further communication with Singular.
  • Martin Albrecht (2006-03-13): added current_ring() and current_ring_name()
  • William Stein (2006-04-10): Fixed problems with ideal constructor
  • Martin Albrecht (2006-05-18): added sage_poly.
  • Simon King (2010-11-23): Reduce the overhead caused by waiting for the Singular prompt by doing garbage collection differently.
  • Simon King (2011-06-06): Make conversion from Singular to Sage more flexible.

Introduction

This interface is extremely flexible, since it’s exactly like typing into the Singular interpreter, and anything that works there should work here.

The Singular interface will only work if Singular is installed on your computer; this should be the case, since Singular is included with Sage. The interface offers three pieces of functionality:

  1. singular_console() - A function that dumps you into an interactive command-line Singular session.
  2. singular(expr, type='def') - Creation of a Singular object. This provides a Pythonic interface to Singular. For example, if f=singular(10), then f.factorize() returns the factorization of \(10\) computed using Singular.
  3. singular.eval(expr) - Evaluation of arbitrary Singular expressions, with the result returned as a string.

Of course, there are polynomial rings and ideals in Sage as well (often based on a C-library interface to Singular). One can convert an object in the Singular interpreter interface to Sage by the method sage().

Tutorial

EXAMPLES: First we illustrate multivariate polynomial factorization:

sage: R1 = singular.ring(0, '(x,y)', 'dp')
sage: R1
//   characteristic : 0
//   number of vars : 2
//        block   1 : ordering dp
//                  : names    x y
//        block   2 : ordering C
sage: f = singular('9x16 - 18x13y2 - 9x12y3 + 9x10y4 - 18x11y2 + 36x8y4 + 18x7y5 - 18x5y6 + 9x6y4 - 18x3y6 - 9x2y7 + 9y8')
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]
9,
x^6-2*x^3*y^2-x^2*y^3+y^4,
-x^5+y^2
sage: F[1][2]
x^6-2*x^3*y^2-x^2*y^3+y^4

We can convert \(f\) and each exponent back to Sage objects as well.

sage: g = f.sage(); g
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[1][2].sage()
x^6 - 2*x^3*y^2 - x^2*y^3 + y^4
sage: g.parent()
Multivariate Polynomial Ring in x, y over Rational Field

This example illustrates polynomial GCD’s:

sage: R2 = singular.ring(0, '(x,y,z)', 'lp')
sage: a = singular.new('3x2*(x+y)')
sage: b = singular.new('9x*(y2-x2)')
sage: g = a.gcd(b)
sage: g
x^2+x*y

This example illustrates computation of a Groebner basis:

sage: R3 = singular.ring(0, '(a,b,c,d)', 'lp')
sage: I = singular.ideal(['a + b + c + d', 'a*b + a*d + b*c + c*d', 'a*b*c + a*b*d + a*c*d + b*c*d', 'a*b*c*d - 1'])
sage: I2 = I.groebner()
sage: I2
c^2*d^6-c^2*d^2-d^4+1,
c^3*d^2+c^2*d^3-c-d,
b*d^4-b+d^5-d,
b*c-b*d^5+c^2*d^4+c*d-d^6-d^2,
b^2+2*b*d+d^2,
a+b+c+d

The following example is the same as the one in the Singular - Gap interface documentation:

sage: R  = singular.ring(0, '(x0,x1,x2)', 'lp')
sage: I1 = singular.ideal(['x0*x1*x2 -x0^2*x2', 'x0^2*x1*x2-x0*x1^2*x2-x0*x1*x2^2', 'x0*x1-x0*x2-x1*x2'])
sage: I2 = I1.groebner()
sage: I2
x1^2*x2^2,
x0*x2^3-x1^2*x2^2+x1*x2^3,
x0*x1-x0*x2-x1*x2,
x0^2*x2-x0*x2^2-x1*x2^2
sage: I2.sage()
Ideal (x1^2*x2^2, x0*x2^3 - x1^2*x2^2 + x1*x2^3, x0*x1 - x0*x2 - x1*x2, x0^2*x2 - x0*x2^2 - x1*x2^2) of Multivariate Polynomial Ring in x0, x1, x2 over Rational Field

This example illustrates moving a polynomial from one ring to another. It also illustrates calling a method of an object with an argument.

sage: R = singular.ring(0, '(x,y,z)', 'dp')
sage: f = singular('x3+y3+(x-y)*x2y2+z2')
sage: f
x^3*y^2-x^2*y^3+x^3+y^3+z^2
sage: R1 = singular.ring(0, '(x,y,z)', 'ds')
sage: f = R.fetch(f)
sage: f
z^2+x^3+y^3+x^3*y^2-x^2*y^3

We can calculate the Milnor number of \(f\):

sage: _=singular.LIB('sing.lib')     # assign to _ to suppress printing
sage: f.milnor()
4

The Jacobian applied twice yields the Hessian matrix of \(f\), with which we can compute.

sage: H = f.jacob().jacob()
sage: H
6*x+6*x*y^2-2*y^3,6*x^2*y-6*x*y^2,  0,
6*x^2*y-6*x*y^2,  6*y+2*x^3-6*x^2*y,0,
0,                0,                2
sage: H.sage()
[6*x + 6*x*y^2 - 2*y^3     6*x^2*y - 6*x*y^2                     0]
[    6*x^2*y - 6*x*y^2 6*y + 2*x^3 - 6*x^2*y                     0]
[                    0                     0                     2]
sage: H.det()   # This is a polynomial in Singular
72*x*y+24*x^4-72*x^3*y+72*x*y^3-24*y^4-48*x^4*y^2+64*x^3*y^3-48*x^2*y^4
sage: H.det().sage()   # This is the corresponding polynomial in Sage
72*x*y + 24*x^4 - 72*x^3*y + 72*x*y^3 - 24*y^4 - 48*x^4*y^2 + 64*x^3*y^3 - 48*x^2*y^4

The 1x1 and 2x2 minors:

sage: H.minor(1)
2,
6*y+2*x^3-6*x^2*y,
6*x^2*y-6*x*y^2,
6*x^2*y-6*x*y^2,
6*x+6*x*y^2-2*y^3
sage: H.minor(2)
12*y+4*x^3-12*x^2*y,
12*x^2*y-12*x*y^2,
12*x^2*y-12*x*y^2,
12*x+12*x*y^2-4*y^3,
-36*x*y-12*x^4+36*x^3*y-36*x*y^3+12*y^4+24*x^4*y^2-32*x^3*y^3+24*x^2*y^4
sage: _=singular.eval('option(redSB)')
sage: H.minor(1).groebner()
1

Computing the Genus

We compute the projective genus of ideals that define curves over \(\QQ\). It is very important to load the normal.lib library before calling the genus command, or you’ll get an error message.

EXAMPLE:

sage: singular.lib('normal.lib')
sage: R = singular.ring(0,'(x,y)','dp')
sage: i2 = singular.ideal('y9 - x2*(x-1)^9 + x')
sage: i2.genus()
40

Note that the genus can be much smaller than the degree:

sage: i = singular.ideal('y9 - x2*(x-1)^9')
sage: i.genus()
0

An Important Concept

AUTHORS:

  • Neal Harris

The following illustrates an important concept: how Sage interacts with the data being used and returned by Singular. Let’s compute a Groebner basis for some ideal, using Singular through Sage.

sage: singular.lib('poly.lib')
sage: singular.ring(32003, '(a,b,c,d,e,f)', 'lp')
        //   characteristic : 32003
        //   number of vars : 6
        //        block   1 : ordering lp
        //                        : names    a b c d e f
        //        block   2 : ordering C
sage: I = singular.ideal('cyclic(6)')
sage: g = singular('groebner(I)')
Traceback (most recent call last):
...
TypeError: Singular error:
...

We restart everything and try again, but correctly.

sage: singular.quit()
sage: singular.lib('poly.lib'); R = singular.ring(32003, '(a,b,c,d,e,f)', 'lp')
sage: I = singular.ideal('cyclic(6)')
sage: I.groebner()
f^48-2554*f^42-15674*f^36+12326*f^30-12326*f^18+15674*f^12+2554*f^6-1,
...

It’s important to understand why the first attempt at computing a basis failed. The line where we gave singular the input ‘groebner(I)’ was useless because Singular has no idea what ‘I’ is! Although ‘I’ is an object that we computed with calls to Singular functions, it actually lives in Sage. As a consequence, the name ‘I’ means nothing to Singular. When we called I.groebner(), Sage was able to call the groebner function on’I’ in Singular, since ‘I’ actually means something to Sage.

Long Input

The Singular interface reads in even very long input (using files) in a robust manner, as long as you are creating a new object.

sage: t = '"%s"'%10^15000   # 15 thousand character string (note that normal Singular input must be at most 10000)
sage: a = singular.eval(t)
sage: a = singular(t)

TESTS:

We test an automatic coercion:

sage: a = 3*singular('2'); a
6
sage: type(a)
<class 'sage.interfaces.singular.SingularElement'>
sage: a = singular('2')*3; a
6
sage: type(a)
<class 'sage.interfaces.singular.SingularElement'>

Create a ring over GF(9) to check that gftables has been installed, see ticket #11645:

sage: singular.eval("ring testgf9 = (9,x),(a,b,c,d,e,f),(M((1,2,3,0)),wp(2,3),lp);")
'ring testgf9 = (9,x),(a,b,c,d,e,f),(M((1,2,3,0)),wp(2,3),lp);'
class sage.interfaces.singular.Singular(maxread=1000, script_subdirectory=None, logfile=None, server=None, server_tmpdir=None)

Bases: sage.interfaces.expect.Expect

Interface to the Singular interpreter.

EXAMPLES: A Groebner basis example.

sage: R = singular.ring(0, '(x0,x1,x2)', 'lp')
sage: I = singular.ideal([ 'x0*x1*x2 -x0^2*x2', 'x0^2*x1*x2-x0*x1^2*x2-x0*x1*x2^2', 'x0*x1-x0*x2-x1*x2'])
sage: I.groebner()
x1^2*x2^2,
x0*x2^3-x1^2*x2^2+x1*x2^3,
x0*x1-x0*x2-x1*x2,
x0^2*x2-x0*x2^2-x1*x2^2

AUTHORS:

  • David Joyner and William Stein
LIB(lib, reload=False)

Load the Singular library named lib.

Note that if the library was already loaded during this session it is not reloaded unless the optional reload argument is True (the default is False).

EXAMPLES:

sage: singular.lib('sing.lib')
sage: singular.lib('sing.lib', reload=True)
clear(var)

Clear the variable named var.

EXAMPLES:

sage: singular.set('int', 'x', '2')
sage: singular.get('x')
'2'
sage: singular.clear('x')

“Clearing the variable” means to allow to free the memory that it uses in the Singular sub-process. However, the actual deletion of the variable is only committed when the next element in the Singular interface is created:

sage: singular.get('x')
'2'
sage: a = singular(3)
sage: singular.get('x')
'`x`'
console()

EXAMPLES:

sage: singular_console() #not tested
                     SINGULAR                             /  Development
 A Computer Algebra System for Polynomial Computations   /   version 3-0-4
                                                       0<
     by: G.-M. Greuel, G. Pfister, H. Schoenemann        \   Nov 2007
FB Mathematik der Universitaet, D-67653 Kaiserslautern            
cputime(t=None)

Returns the amount of CPU time that the Singular session has used. If t is not None, then it returns the difference between the current CPU time and t.

EXAMPLES:

sage: t = singular.cputime()
sage: R = singular.ring(0, '(x0,x1,x2)', 'lp')
sage: I = singular.ideal([ 'x0*x1*x2 -x0^2*x2', 'x0^2*x1*x2-x0*x1^2*x2-x0*x1*x2^2', 'x0*x1-x0*x2-x1*x2'])
sage: gb = I.groebner()
sage: singular.cputime(t) #random
0.02
current_ring()

Returns the current ring of the running Singular session.

EXAMPLES:

sage: r = PolynomialRing(GF(127),3,'xyz', order='invlex')
sage: r._singular_()
//   characteristic : 127
//   number of vars : 3
//        block   1 : ordering rp
//                  : names    x y z
//        block   2 : ordering C
sage: singular.current_ring()
//   characteristic : 127
//   number of vars : 3
//        block   1 : ordering rp
//                  : names    x y z
//        block   2 : ordering C
current_ring_name()

Returns the Singular name of the currently active ring in Singular.

OUTPUT: currently active ring’s name

EXAMPLES:

sage: r = PolynomialRing(GF(127),3,'xyz')
sage: r._singular_().name() == singular.current_ring_name()
True
eval(x, allow_semicolon=True, strip=True, **kwds)

Send the code x to the Singular interpreter and return the output as a string.

INPUT:

  • x - string (of code)
  • allow_semicolon - default: False; if False then raise a TypeError if the input line contains a semicolon.
  • strip - ignored

EXAMPLES:

sage: singular.eval('2 > 1')
'1'
sage: singular.eval('2 + 2')
'4'

if the verbosity level is \(> 1\) comments are also printed and not only returned.

sage: r = singular.ring(0,'(x,y,z)','dp')
sage: i = singular.ideal(['x^2','y^2','z^2'])
sage: s = i.std()
sage: singular.eval('hilb(%s)'%(s.name()))
'// 1 t^0\n// -3 t^2\n// 3 t^4\n// -1 t^6\n\n// 1 t^0\n//
3 t^1\n// 3 t^2\n// 1 t^3\n// dimension (affine) = 0\n//
degree (affine) = 8'
sage: set_verbose(1)
sage: o = singular.eval('hilb(%s)'%(s.name()))
//         1 t^0
//        -3 t^2
//         3 t^4
//        -1 t^6
//         1 t^0
//         3 t^1
//         3 t^2
//         1 t^3
// dimension (affine) = 0
// degree (affine)  = 8

This is mainly useful if this method is called implicitly. Because then intermediate results, debugging outputs and printed statements are printed

sage: o = s.hilb()
//         1 t^0
//        -3 t^2
//         3 t^4
//        -1 t^6
//         1 t^0
//         3 t^1
//         3 t^2
//         1 t^3
// dimension (affine) = 0
// degree (affine)  = 8
// ** right side is not a datum, assignment ignored

rather than ignored

sage: set_verbose(0)
sage: o = s.hilb()
get(var)

Get string representation of variable named var.

EXAMPLES:

sage: singular.set('int', 'x', '2')
sage: singular.get('x')
'2'
has_coerce_map_from_impl(S)

x.__init__(...) initializes x; see help(type(x)) for signature

ideal(*gens)

Return the ideal generated by gens.

INPUT:

  • gens - list or tuple of Singular objects (or objects that can be made into Singular objects via evaluation)

OUTPUT: the Singular ideal generated by the given list of gens

EXAMPLES: A Groebner basis example done in a different way.

sage: _ = singular.eval("ring R=0,(x0,x1,x2),lp")
sage: i1 = singular.ideal([ 'x0*x1*x2 -x0^2*x2', 'x0^2*x1*x2-x0*x1^2*x2-x0*x1*x2^2', 'x0*x1-x0*x2-x1*x2'])
sage: i1
-x0^2*x2+x0*x1*x2,
x0^2*x1*x2-x0*x1^2*x2-x0*x1*x2^2,
x0*x1-x0*x2-x1*x2
sage: i2 = singular.ideal('groebner(%s);'%i1.name())
sage: i2
x1^2*x2^2,
x0*x2^3-x1^2*x2^2+x1*x2^3,
x0*x1-x0*x2-x1*x2,
x0^2*x2-x0*x2^2-x1*x2^2
lib(lib, reload=False)

Load the Singular library named lib.

Note that if the library was already loaded during this session it is not reloaded unless the optional reload argument is True (the default is False).

EXAMPLES:

sage: singular.lib('sing.lib')
sage: singular.lib('sing.lib', reload=True)
list(x)

Creates a list in Singular from a Sage list x.

EXAMPLES:

sage: singular.list([1,2])
[1]:
   1
[2]:
   2
load(lib, reload=False)

Load the Singular library named lib.

Note that if the library was already loaded during this session it is not reloaded unless the optional reload argument is True (the default is False).

EXAMPLES:

sage: singular.lib('sing.lib')
sage: singular.lib('sing.lib', reload=True)
matrix(nrows, ncols, entries=None)

EXAMPLES:

sage: singular.lib("matrix")
sage: R = singular.ring(0, '(x,y,z)', 'dp')
sage: A = singular.matrix(3,2,'1,2,3,4,5,6')
sage: A
1,2,
3,4,
5,6
sage: A.gauss_col()
2,-1,
1,0,
0,1

AUTHORS:

  • Martin Albrecht (2006-01-14)
option(cmd=None, val=None)

Access to Singular’s options as follows:

Syntax: option() Returns a string of all defined options.

Syntax: option( ‘option_name’ ) Sets an option. Note to disable an option, use the prefix no.

Syntax: option( ‘get’ ) Returns an intvec of the state of all options.

Syntax: option( ‘set’, intvec_expression ) Restores the state of all options from an intvec (produced by option(‘get’)).

EXAMPLES:

sage: singular.option()
//options: redefine loadLib usage prompt
sage: singular.option('get')
0,
10321
sage: old_options = _
sage: singular.option('noredefine')
sage: singular.option()
//options: loadLib usage prompt
sage: singular.option('set', old_options)
sage: singular.option('get')
0,
10321
ring(char=0, vars='(x)', order='lp', check=True)

Create a Singular ring and makes it the current ring.

INPUT:

  • char - characteristic of the base ring (see examples below), which must be either 0, prime (!), or one of several special codes (see examples below).
  • vars - a tuple or string that defines the variable names
  • order - string - the monomial order (default: ‘lp’)
  • check - if True, check primality of the characteristic if it is an integer.

OUTPUT: a Singular ring

Note

This function is not identical to calling the Singular ring function. In particular, it also attempts to “kill” the variable names, so they can actually be used without getting errors, and it sets printing of elements for this range to short (i.e., with *’s and carets).

EXAMPLES: We first declare \(\QQ[x,y,z]\) with degree reverse lexicographic ordering.

sage: R = singular.ring(0, '(x,y,z)', 'dp')
sage: R
//   characteristic : 0
//   number of vars : 3
//        block   1 : ordering dp
//                  : names    x y z
//        block   2 : ordering C
sage: R1 = singular.ring(32003, '(x,y,z)', 'dp')
sage: R2 = singular.ring(32003, '(a,b,c,d)', 'lp')

This is a ring in variables named x(1) through x(10) over the finite field of order \(7\):

sage: R3 = singular.ring(7, '(x(1..10))', 'ds')

This is a polynomial ring over the transcendental extension \(\QQ(a)\) of \(\QQ\):

sage: R4 = singular.ring('(0,a)', '(mu,nu)', 'lp')

This is a ring over the field of single-precision floats:

sage: R5 = singular.ring('real', '(a,b)', 'lp')

This is over 50-digit floats:

sage: R6 = singular.ring('(real,50)', '(a,b)', 'lp')
sage: R7 = singular.ring('(complex,50,i)', '(a,b)', 'lp')

To use a ring that you’ve defined, use the set_ring() method on the ring. This sets the ring to be the “current ring”. For example,

sage: R = singular.ring(7, '(a,b)', 'ds')
sage: S = singular.ring('real', '(a,b)', 'lp')
sage: singular.new('10*a')
1.000e+01*a
sage: R.set_ring()
sage: singular.new('10*a')
3*a
set(type, name, value)

Set the variable with given name to the given value.

REMARK:

If a variable in the Singular interface was previously marked for deletion, the actual deletion is done here, before the new variable is created in Singular.

EXAMPLES:

sage: singular.set('int', 'x', '2')
sage: singular.get('x')
'2'

We test that an unused variable is only actually deleted if this method is called:

sage: a = singular(3)
sage: n = a.name()
sage: del a
sage: singular.eval(n)
'3'
sage: singular.set('int', 'y', '5')
sage: singular.eval('defined(%s)'%n)
'0'
set_ring(R)

Sets the current Singular ring to R.

EXAMPLES:

sage: R = singular.ring(7, '(a,b)', 'ds')
sage: S = singular.ring('real', '(a,b)', 'lp')
sage: singular.current_ring()
//   characteristic : 0 (real)
//   number of vars : 2
//        block   1 : ordering lp
//                  : names    a b
//        block   2 : ordering C
sage: singular.set_ring(R)
sage: singular.current_ring()
//   characteristic : 7
//   number of vars : 2
//        block   1 : ordering ds
//                  : names    a b
//        block   2 : ordering C
setring(R)

Sets the current Singular ring to R.

EXAMPLES:

sage: R = singular.ring(7, '(a,b)', 'ds')
sage: S = singular.ring('real', '(a,b)', 'lp')
sage: singular.current_ring()
//   characteristic : 0 (real)
//   number of vars : 2
//        block   1 : ordering lp
//                  : names    a b
//        block   2 : ordering C
sage: singular.set_ring(R)
sage: singular.current_ring()
//   characteristic : 7
//   number of vars : 2
//        block   1 : ordering ds
//                  : names    a b
//        block   2 : ordering C
string(x)

Creates a Singular string from a Sage string. Note that the Sage string has to be “double-quoted”.

EXAMPLES:

sage: singular.string('"Sage"')
Sage
trait_names()

Return a list of all Singular commands.

EXAMPLES:

sage: singular.trait_names()
['exteriorPower',
 ...
 'stdfglm']
version()

EXAMPLES:

class sage.interfaces.singular.SingularElement(parent, type, value, is_name=False)

Bases: sage.interfaces.expect.ExpectElement

EXAMPLES:

sage: a = singular(2)
sage: loads(dumps(a))
(invalid object -- defined in terms of closed session)
attrib(name, value=None)

Get and set attributes for self.

INPUT:

  • name - string to choose the attribute
  • value - boolean value or None for reading, (default:None)

VALUES: isSB - the standard basis property is set by all commands computing a standard basis like groebner, std, stdhilb etc.; used by lift, dim, degree, mult, hilb, vdim, kbase isHomog - the weight vector for homogeneous or quasihomogeneous ideals/modules isCI - complete intersection property isCM - Cohen-Macaulay property rank - set the rank of a module (see nrows) withSB - value of type ideal, resp. module, is std withHilb - value of type intvec is hilb(_,1) (see hilb) withRes - value of type list is a free resolution withDim - value of type int is the dimension (see dim) withMult - value of type int is the multiplicity (see mult)

EXAMPLE:

sage: P.<x,y,z> = PolynomialRing(QQ)
sage: I = Ideal([z^2, y*z, y^2, x*z, x*y, x^2])
sage: Ibar = I._singular_()
sage: Ibar.attrib('isSB')
0
sage: singular.eval('vdim(%s)'%Ibar.name()) # sage7 name is random
// ** sage7 is no standard basis
4
sage: Ibar.attrib('isSB',1)
sage: singular.eval('vdim(%s)'%Ibar.name())
'4'
sage_flattened_str_list()

EXAMPLES:

sage: R=singular.ring(0,'(x,y)','dp')
sage: RL = R.ringlist()
sage: RL.sage_flattened_str_list()
['0', 'x', 'y', 'dp', '1,1', 'C', '0', '_[1]=0']
sage_global_ring()

Return the current basering in Singular as a polynomial ring or quotient ring.

EXAMPLE:

sage: singular.eval('ring r1 = (9,x),(a,b,c,d,e,f),(M((1,2,3,0)),wp(2,3),lp)')
'ring r1 = (9,x),(a,b,c,d,e,f),(M((1,2,3,0)),wp(2,3),lp);'
sage: R = singular('r1').sage_global_ring()
sage: R
Multivariate Polynomial Ring in a, b, c, d, e, f over Finite Field in x of size 3^2
sage: R.term_order()
Block term order with blocks:
(Matrix term order with matrix
[1 2]
[3 0],
 Weighted degree reverse lexicographic term order with weights (2, 3),
 Lexicographic term order of length 2)
sage: singular.eval('ring r2 = (0,x),(a,b,c),dp')
'ring r2 = (0,x),(a,b,c),dp;'
sage: singular('r2').sage_global_ring()
Multivariate Polynomial Ring in a, b, c over Fraction Field of Univariate Polynomial Ring in x over Rational Field
sage: singular.eval('ring r3 = (3,z),(a,b,c),dp')
'ring r3 = (3,z),(a,b,c),dp;'
sage: singular.eval('minpoly = 1+z+z2+z3+z4')
'minpoly = 1+z+z2+z3+z4;'
sage: singular('r3').sage_global_ring()
Multivariate Polynomial Ring in a, b, c over Finite Field in z of size 3^4

Real and complex fields in both Singular and Sage are defined with a precision. The precision in Singular is given in terms of digits, but in Sage it is given in terms of bits. So, the digit precision is internally converted to a reasonable bit precision:

sage: singular.eval('ring r4 = (real,20),(a,b,c),dp')
'ring r4 = (real,20),(a,b,c),dp;'
sage: singular('r4').sage_global_ring()
Multivariate Polynomial Ring in a, b, c over Real Field with 70 bits of precision

The case of complex coefficients is not fully supported, yet, since the generator of a complex field in Sage is always called “I”:

sage: singular.eval('ring r5 = (complex,15,j),(a,b,c),dp')
'ring r5 = (complex,15,j),(a,b,c),dp;'
sage: R = singular('r5').sage_global_ring(); R
Multivariate Polynomial Ring in a, b, c over Complex Field with 54 bits of precision
sage: R.base_ring()('j')
Traceback (most recent call last):
...
NameError: name 'j' is not defined
sage: R.base_ring()('I')
1.00000000000000*I

In our last example, the base ring is a quotient ring:

sage: singular.eval('ring r6 = (9,a), (x,y,z),lp')
'ring r6 = (9,a), (x,y,z),lp;'
sage: Q = singular('std(ideal(x^2,x+y^2+z^3))', type='qring')
sage: Q.sage_global_ring()
Quotient of Multivariate Polynomial Ring in x, y, z over Finite Field in a of size 3^2 by the ideal (y^4 - y^2*z^3 + z^6, x + y^2 + z^3)

AUTHOR:

  • Simon King (2011-06-06)
sage_matrix(R, sparse=True)

Returns Sage matrix for self

INPUT:

  • R - (default: None); an optional ring, over which the resulting matrix is going to be defined. By default, the output of sage_global_ring() is used.
  • sparse - (default: True); determines whether the resulting matrix is sparse or not.

EXAMPLES:

sage: R = singular.ring(0, '(x,y,z)', 'dp')
sage: A = singular.matrix(2,2)
sage: A.sage_matrix(ZZ)
[0 0]
[0 0]
sage: A.sage_matrix(RDF)
[0.0 0.0]
[0.0 0.0]
sage_poly(R=None, kcache=None)

Returns a Sage polynomial in the ring r matching the provided poly which is a singular polynomial.

INPUT:

  • R - (default: None); an optional polynomial ring. If it is provided, then you have to make sure that it matches the current singular ring as, e.g., returned by singular.current_ring(). By default, the output of sage_global_ring() is used.
  • kcache - (default: None); an optional dictionary for faster finite field lookups, this is mainly useful for finite extension fields

OUTPUT: MPolynomial

EXAMPLES:

sage: R = PolynomialRing(GF(2^8,'a'),2,'xy')
sage: f=R('a^20*x^2*y+a^10+x')
sage: f._singular_().sage_poly(R)==f
True
sage: R = PolynomialRing(GF(2^8,'a'),1,'x')
sage: f=R('a^20*x^3+x^2+a^10')
sage: f._singular_().sage_poly(R)==f
True
sage: P.<x,y> = PolynomialRing(QQ, 2)
sage: f = x*y**3 - 1/9 * x + 1; f
x*y^3 - 1/9*x + 1
sage: singular(f)
x*y^3-1/9*x+1
sage: P(singular(f))
x*y^3 - 1/9*x + 1

TESTS:

sage: singular.eval('ring r = (3,z),(a,b,c),dp')
'ring r = (3,z),(a,b,c),dp;'
sage: singular.eval('minpoly = 1+z+z2+z3+z4')
'minpoly = 1+z+z2+z3+z4;'
sage: p = singular('z^4*a^3+z^2*a*b*c')
sage: p.sage_poly()
(-z^3 - z^2 - z - 1)*a^3 + (z^2)*a*b*c
sage: singular('z^4')
(-z3-z2-z-1)

AUTHORS:

  • Martin Albrecht (2006-05-18)
  • Simon King (2011-06-06): Deal with Singular’s short polynomial representation, automatic construction of a polynomial ring, if it is not explicitly given.

Note

For very simple polynomials eval(SingularElement.sage_polystring()) is faster than SingularElement.sage_poly(R), maybe we should detect the crossover point (in dependence of the string length) and choose an appropriate conversion strategy

sage_polystring()

If this Singular element is a polynomial, return a string representation of this polynomial that is suitable for evaluation in Python. Thus * is used for multiplication and ** for exponentiation. This function is primarily used internally.

The short=0 option must be set for the parent ring or this function will not work as expected. This option is set by default for rings created using singular.ring or set using ring_name.set_ring().

EXAMPLES:

sage: R = singular.ring(0,'(x,y)')
sage: f = singular('x^3 + 3*y^11 + 5')
sage: f
x^3+3*y^11+5
sage: f.sage_polystring()
'x**3+3*y**11+5'
sage_structured_str_list()

If self is a Singular list of lists of Singular elements, returns corresponding Sage list of lists of strings.

EXAMPLES:

sage: R=singular.ring(0,'(x,y)','dp')
sage: RL=R.ringlist()
sage: RL
[1]:
   0
[2]:
   [1]:
      x
   [2]:
      y
[3]:
   [1]:
      [1]:
         dp
      [2]:
         1,1
   [2]:
      [1]:
         C
      [2]:
         0
[4]:
   _[1]=0
sage: RL.sage_structured_str_list()
['0', ['x', 'y'], [['dp', '1,\n1 '], ['C', '0 ']], '0']
set_ring()

Sets the current ring in Singular to be self.

EXAMPLES:

sage: R = singular.ring(7, '(a,b)', 'ds')
sage: S = singular.ring('real', '(a,b)', 'lp')
sage: singular.current_ring()
//   characteristic : 0 (real)
//   number of vars : 2
//        block   1 : ordering lp
//                  : names    a b
//        block   2 : ordering C
sage: R.set_ring()
sage: singular.current_ring()
//   characteristic : 7
//   number of vars : 2
//        block   1 : ordering ds
//                  : names    a b
//        block   2 : ordering C
trait_names()

Returns the possible tab-completions for self. In this case, we just return all the tab completions for the Singular object.

EXAMPLES:

sage: R = singular.ring(0,'(x,y)','dp')
sage: R.trait_names()
['exteriorPower',
 ...
 'stdfglm']
type()

Returns the internal type of this element.

EXAMPLES:

sage: R = PolynomialRing(GF(2^8,'a'),2,'x')
sage: R._singular_().type()
'ring'
sage: fs = singular('x0^2','poly')
sage: fs.type()
'poly'
exception sage.interfaces.singular.SingularError

Bases: exceptions.RuntimeError

Raised if Singular printed an error message

class sage.interfaces.singular.SingularFunction(parent, name)

Bases: sage.interfaces.expect.ExpectFunction

class sage.interfaces.singular.SingularFunctionElement(obj, name)

Bases: sage.interfaces.expect.FunctionElement

class sage.interfaces.singular.SingularGBLogPrettyPrinter(verbosity=1)

A device which prints Singular Groebner basis computation logs more verbatim.

flush()

EXAMPLE:

sage: from sage.interfaces.singular import SingularGBLogPrettyPrinter
sage: s3 = SingularGBLogPrettyPrinter(verbosity=3)
sage: s3.flush()
write(s)

EXAMPLE:

sage: from sage.interfaces.singular import SingularGBLogPrettyPrinter
sage: s3 = SingularGBLogPrettyPrinter(verbosity=3)
sage: s3.write("(S:1337)")
Performing complete reduction of 1337 elements.
sage: s3.write("M[389,12]")
Parallel reduction of 389 elements with 12 non-zero output elements.
sage.interfaces.singular.generate_docstring_dictionary()

Generate global dictionaries which hold the docstrings for Singular functions.

EXAMPLE:

sage: from sage.interfaces.singular import generate_docstring_dictionary
sage: generate_docstring_dictionary()
sage.interfaces.singular.get_docstring(name)

Return the docstring for the function name.

INPUT:

  • name - a Singular function name

EXAMPLE:

sage: from sage.interfaces.singular import get_docstring
sage: 'groebner' in get_docstring('groebner')
True
sage: 'standard.lib' in get_docstring('groebner')
True
sage.interfaces.singular.is_SingularElement(x)

Returns True is x is of type SingularElement.

EXAMPLES:

sage: from sage.interfaces.singular import is_SingularElement
sage: is_SingularElement(singular(2))
True
sage: is_SingularElement(2)
False
sage.interfaces.singular.reduce_load()

Note that this returns an invalid Singular object!

EXAMPLES:

sage: from sage.interfaces.singular import reduce_load
sage: reduce_load()
(invalid object -- defined in terms of closed session)
sage.interfaces.singular.reduce_load_Singular()

EXAMPLES:

sage: from sage.interfaces.singular import reduce_load_Singular
sage: reduce_load_Singular()
Singular
sage.interfaces.singular.singular_console()

Spawn a new Singular command-line session.

EXAMPLES:

sage: singular_console() #not tested
                     SINGULAR                             /  Development
 A Computer Algebra System for Polynomial Computations   /   version 3-0-4
                                                       0<
     by: G.-M. Greuel, G. Pfister, H. Schoenemann        \   Nov 2007
FB Mathematik der Universitaet, D-67653 Kaiserslautern        
sage.interfaces.singular.singular_version()

Returns the version of Singular being used.

EXAMPLES:

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