# libSingular: Functions#

Sage implements a C wrapper around the Singular interpreter which allows to call any function directly from Sage without string parsing or interprocess communication overhead. Users who do not want to call Singular functions directly, usually do not have to worry about this interface, since it is handled by higher level functions in Sage.

EXAMPLES:

The direct approach for loading a Singular function is to call the function `singular_function()` with the function name as parameter:

```sage: from sage.libs.singular.function import singular_function
sage: P.<a,b,c,d> = PolynomialRing(GF(7))
sage: std = singular_function('std')
sage: I = sage.rings.ideal.Cyclic(P)
sage: std(I)
[a + b + c + d,
b^2 + 2*b*d + d^2,
b*c^2 + c^2*d - b*d^2 - d^3,
b*c*d^2 + c^2*d^2 - b*d^3 + c*d^3 - d^4 - 1,
b*d^4 + d^5 - b - d,
c^3*d^2 + c^2*d^3 - c - d,
c^2*d^4 + b*c - b*d + c*d - 2*d^2]
```

If a Singular library needs to be loaded before a certain function is available, use the `lib()` function as shown below:

```sage: from sage.libs.singular.function import singular_function, lib as singular_lib
sage: primdecSY = singular_function('primdecSY')
Traceback (most recent call last):
...
NameError: Singular library function 'primdecSY' is not defined

sage: singular_lib('primdec.lib')
sage: primdecSY = singular_function('primdecSY')
```

There is also a short-hand notation for the above:

```sage: import sage.libs.singular.function_factory
sage: primdecSY = sage.libs.singular.function_factory.ff.primdec__lib.primdecSY
```

The above line will load “primdec.lib” first and then load the function `primdecSY`.

AUTHORS:

• Michael Brickenstein (2009-07): initial implementation, overall design

• Martin Albrecht (2009-07): clean up, enhancements, etc

• Michael Brickenstein (2009-10): extension to more Singular types

• Martin Albrecht (2010-01): clean up, support for attributes

• Simon King (2011-04): include the documentation provided by Singular as a code block

• Burcin Erocal, Michael Brickenstein, Oleksandr Motsak, Alexander Dreyer, Simon King (2011-09): plural support

class sage.libs.singular.function.BaseCallHandler#

Bases: `object`

A call handler is an abstraction which hides the details of the implementation differences between kernel and library functions.

class sage.libs.singular.function.Converter#

Bases: `SageObject`

A `Converter` interfaces between Sage objects and Singular interpreter objects.

ring()#

Return the ring in which the arguments of this list live.

EXAMPLES:

```sage: from sage.libs.singular.function import Converter
sage: P.<a,b,c> = PolynomialRing(GF(127))
sage: Converter([a,b,c],ring=P).ring()
Multivariate Polynomial Ring in a, b, c over Finite Field of size 127
```
class sage.libs.singular.function.KernelCallHandler#

A call handler is an abstraction which hides the details of the implementation differences between kernel and library functions.

This class implements calling a kernel function.

Note

Do not construct this class directly, use `singular_function()` instead.

class sage.libs.singular.function.LibraryCallHandler#

A call handler is an abstraction which hides the details of the implementation differences between kernel and library functions.

This class implements calling a library function.

Note

Do not construct this class directly, use `singular_function()` instead.

class sage.libs.singular.function.Resolution#

Bases: `object`

A simple wrapper around Singular’s resolutions.

class sage.libs.singular.function.RingWrap#

Bases: `object`

A simple wrapper around Singular’s rings.

characteristic()#

Get characteristic.

EXAMPLES:

```sage: from sage.libs.singular.function import singular_function
sage: P.<x,y,z> = PolynomialRing(QQ)
sage: ringlist = singular_function("ringlist")
sage: l = ringlist(P)
sage: ring = singular_function("ring")
sage: ring(l, ring=P).characteristic()
0
```
is_commutative()#

Determine whether a given ring is commutative.

EXAMPLES:

```sage: from sage.libs.singular.function import singular_function
sage: P.<x,y,z> = PolynomialRing(QQ)
sage: ringlist = singular_function("ringlist")
sage: l = ringlist(P)
sage: ring = singular_function("ring")
sage: ring(l, ring=P).is_commutative()
True
```
ngens()#

Get number of generators.

EXAMPLES:

```sage: from sage.libs.singular.function import singular_function
sage: P.<x,y,z> = PolynomialRing(QQ)
sage: ringlist = singular_function("ringlist")
sage: l = ringlist(P)
sage: ring = singular_function("ring")
sage: ring(l, ring=P).ngens()
3
```
npars()#

Get number of parameters.

EXAMPLES:

```sage: from sage.libs.singular.function import singular_function
sage: P.<x,y,z> = PolynomialRing(QQ)
sage: ringlist = singular_function("ringlist")
sage: l = ringlist(P)
sage: ring = singular_function("ring")
sage: ring(l, ring=P).npars()
0
```
ordering_string()#

Get Singular string defining monomial ordering.

EXAMPLES:

```sage: from sage.libs.singular.function import singular_function
sage: P.<x,y,z> = PolynomialRing(QQ)
sage: ringlist = singular_function("ringlist")
sage: l = ringlist(P)
sage: ring = singular_function("ring")
sage: ring(l, ring=P).ordering_string()
'dp(3),C'
```
par_names()#

Get parameter names.

EXAMPLES:

```sage: from sage.libs.singular.function import singular_function
sage: P.<x,y,z> = PolynomialRing(QQ)
sage: ringlist = singular_function("ringlist")
sage: l = ringlist(P)
sage: ring = singular_function("ring")
sage: ring(l, ring=P).par_names()
[]
```
var_names()#

Get names of variables.

EXAMPLES:

```sage: from sage.libs.singular.function import singular_function
sage: P.<x,y,z> = PolynomialRing(QQ)
sage: ringlist = singular_function("ringlist")
sage: l = ringlist(P)
sage: ring = singular_function("ring")
sage: ring(l, ring=P).var_names()
['x', 'y', 'z']
```
class sage.libs.singular.function.SingularFunction#

Bases: `SageObject`

The base class for Singular functions either from the kernel or from the library.

class sage.libs.singular.function.SingularKernelFunction#

EXAMPLES:

```sage: from sage.libs.singular.function import SingularKernelFunction
sage: R.<x,y> = PolynomialRing(QQ, order='lex')
sage: I = R.ideal(x, x+1)
sage: f = SingularKernelFunction("std")
sage: f(I)
[1]
```
class sage.libs.singular.function.SingularLibraryFunction#

EXAMPLES:

```sage: from sage.libs.singular.function import SingularLibraryFunction
sage: R.<x,y> = PolynomialRing(QQ, order='lex')
sage: I = R.ideal(x, x+1)
sage: f = SingularLibraryFunction("groebner")
sage: f(I)
[1]
```
sage.libs.singular.function.all_singular_poly_wrapper(s)#

Tests for a sequence `s`, whether it consists of singular polynomials.

EXAMPLES:

```sage: from sage.libs.singular.function import all_singular_poly_wrapper
sage: P.<x,y,z> = QQ[]
sage: all_singular_poly_wrapper([x+1, y])
True
sage: all_singular_poly_wrapper([x+1, y, 1])
False
```
sage.libs.singular.function.all_vectors(s)#

Check if a sequence `s` consists of free module elements over a singular ring.

EXAMPLES:

```sage: from sage.libs.singular.function import all_vectors
sage: P.<x,y,z> = QQ[]
sage: M = P**2
sage: all_vectors([x])
False
sage: all_vectors([(x,y)])
False
sage: all_vectors([M(0), M((x,y))])
True
sage: all_vectors([M(0), M((x,y)),(0,0)])
False
```
sage.libs.singular.function.is_sage_wrapper_for_singular_ring(ring)#

Check whether wrapped ring arises from Singular or Singular/Plural.

EXAMPLES:

```sage: from sage.libs.singular.function import is_sage_wrapper_for_singular_ring
sage: P.<x,y,z> = QQ[]
sage: is_sage_wrapper_for_singular_ring(P)
True
```
```sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex')
sage: is_sage_wrapper_for_singular_ring(P)
True
```
sage.libs.singular.function.is_singular_poly_wrapper(p)#

Checks if p is some data type corresponding to some singular `poly`.

EXAMPLES:

```sage: from sage.libs.singular.function import is_singular_poly_wrapper
sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: H.<x,y,z> = A.g_algebra({z*x:x*z+2*x, z*y:y*z-2*y})
sage: is_singular_poly_wrapper(x+y)
True
```
sage.libs.singular.function.lib(name)#

Load the Singular library `name`.

INPUT:

• `name` – a Singular library name

EXAMPLES:

```sage: from sage.libs.singular.function import singular_function
sage: from sage.libs.singular.function import lib as singular_lib
sage: singular_lib('general.lib')
sage: primes = singular_function('primes')
sage: primes(2,10, ring=GF(127)['x,y,z'])
(2, 3, 5, 7)
```
sage.libs.singular.function.list_of_functions(packages=False)#

Return a list of all function names currently available.

INPUT:

• `packages` – include local functions in packages.

EXAMPLES:

```sage: from sage.libs.singular.function import list_of_functions
sage: 'groebner' in list_of_functions()
True
```
sage.libs.singular.function.singular_function(name)#

Construct a new libSingular function object for the given `name`.

This function works both for interpreter and built-in functions.

INPUT:

• `name` – the name of the function

EXAMPLES:

```sage: P.<x,y,z> = PolynomialRing(QQ)
sage: f = 3*x*y + 2*z + 1
sage: g = 2*x + 1/2
sage: I = Ideal([f,g])
```
```sage: from sage.libs.singular.function import singular_function
sage: std = singular_function("std")
sage: std(I)
[3*y - 8*z - 4, 4*x + 1]
sage: size = singular_function("size")
sage: size([2, 3, 3])
3
sage: size("sage")
4
sage: size(["hello", "sage"])
2
sage: factorize = singular_function("factorize")
sage: factorize(f)
[[1, 3*x*y + 2*z + 1], (1, 1)]
sage: factorize(f, 1)
[3*x*y + 2*z + 1]
```

We give a wrong number of arguments:

```sage: factorize()
Traceback (most recent call last):
...
RuntimeError: error in Singular function call 'factorize':
Wrong number of arguments (got 0 arguments, arity code is 305)
sage: factorize(f, 1, 2)
Traceback (most recent call last):
...
RuntimeError: error in Singular function call 'factorize':
Wrong number of arguments (got 3 arguments, arity code is 305)
sage: factorize(f, 1, 2, 3)
Traceback (most recent call last):
...
RuntimeError: error in Singular function call 'factorize':
Wrong number of arguments (got 4 arguments, arity code is 305)
```

The Singular function `list` can be called with any number of arguments:

```sage: singular_list = singular_function("list")
sage: singular_list(2, 3, 6)
[2, 3, 6]
sage: singular_list()
[]
sage: singular_list(1)
[1]
sage: singular_list(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
```

We try to define a non-existing function:

```sage: number_foobar = singular_function('number_foobar')
Traceback (most recent call last):
...
NameError: Singular library function 'number_foobar' is not defined
```
```sage: from sage.libs.singular.function import lib as singular_lib
sage: singular_lib('general.lib')
sage: number_e = singular_function('number_e')
sage: number_e(10r)
67957045707/25000000000
sage: RR(number_e(10r))
2.71828182828000
```
```sage: singular_lib('primdec.lib')
sage: primdecGTZ = singular_function("primdecGTZ")
sage: primdecGTZ(I)
[[[y - 8/3*z - 4/3, x + 1/4], [y - 8/3*z - 4/3, x + 1/4]]]
sage: singular_list((1,2,3),3,[1,2,3], ring=P)
[(1, 2, 3), 3, [1, 2, 3]]
sage: ringlist=singular_function("ringlist")
sage: l = ringlist(P)
sage: l[3].__class__
<class 'sage.rings.polynomial.multi_polynomial_sequence.PolynomialSequence_generic'>
sage: l
[0, ['x', 'y', 'z'], [['dp', (1, 1, 1)], ['C', (0,)]], [0]]
sage: ring=singular_function("ring")
sage: ring(l)
<RingWrap>
sage: matrix = Matrix(P,2,2)
sage: matrix.randomize(terms=1)
sage: det = singular_function("det")
sage: det(matrix) == matrix[0, 0] * matrix[1, 1] - matrix[0, 1] * matrix[1, 0]
True
sage: coeffs = singular_function("coeffs")
sage: coeffs(x*y+y+1,y)
[    1]
[x + 1]
sage: intmat = Matrix(ZZ, 2,2, [100,2,3,4])
sage: det(intmat)
394
sage: random = singular_function("random")
sage: A = random(10,2,3); A.nrows(), max(A.list()) <= 10
(2, True)
sage: P.<x,y,z> = PolynomialRing(QQ)
sage: M=P**3
sage: v=M((100*x,5*y,10*z*x*y))
10
sage: v = M([x+y,x*y+y**3,z])
(0, y^3)
sage: jet = singular_function("jet")
sage: jet(v, 2)
(x + y, x*y, z)
sage: syz = singular_function("syz")
sage: I = P.ideal([x+y,x*y-y, y*2,x**2+1])
sage: M = syz(I)
sage: M
[(-2*y, 2, y + 1, 0), (0, -2, x - 1, 0), (x*y - y, -y + 1, 1, -y), (x^2 + 1, -x - 1, -1, -x)]
sage: singular_lib("mprimdec.lib")
sage: syz(M)
[(-x - 1, y - 1, 2*x, -2*y)]
sage: GTZmod = singular_function("GTZmod")
sage: GTZmod(M)
[[[(-2*y, 2, y + 1, 0), (0, x + 1, 1, -y), (0, -2, x - 1, 0), (x*y - y, -y + 1, 1, -y), (x^2 + 1, 0, 0, -x - y)], [0]]]
sage: mres = singular_function("mres")
sage: resolution = mres(M, 0)
sage: resolution
<Resolution>
sage: singular_list(resolution)
[[(-2*y, 2, y + 1, 0), (0, -2, x - 1, 0), (x*y - y, -y + 1, 1, -y), (x^2 + 1, -x - 1, -1, -x)], [(-x - 1, y - 1, 2*x, -2*y)], [(0)]]

sage: A.<x,y> = FreeAlgebra(QQ, 2)
sage: P.<x,y> = A.g_algebra({y*x:-x*y})
sage: I= Sequence([x*y,x+y], check=False, immutable=True)
sage: twostd = singular_function("twostd")
sage: twostd(I)
[x + y, y^2]
sage: M=syz(I)
doctest...
sage: M
[(x + y, x*y)]
sage: syz(M)
[(0)]
sage: mres(I, 0)
<Resolution>
sage: M=P**3
sage: v=M((100*x,5*y,10*y*x*y))
-10
sage: v = M([x+y,x*y+y**3,x])