Formal sums

AUTHORS:

• David Harvey (2006-09-20): changed FormalSum not to derive from “list” anymore, because that breaks new Element interface

• Nick Alexander (2006-12-06): added test cases.

• William Stein (2006, 2009): wrote the first version in 2006, documented it in 2009.

• Volker Braun (2010-07-19): new-style coercions, documentation added. FormalSums now derives from UniqueRepresentation.

FUNCTIONS:

• FormalSums(ring) – create the module of formal finite sums with coefficients in the given ring

• FormalSum(list of pairs (coeff, number)) – create a formal sum

EXAMPLES:

Sage

sage: A = FormalSum([(1, 2/3)]); A
2/3
sage: B = FormalSum([(3, 1/5)]); B
3*1/5
sage: -B
-3*1/5
sage: A + B
2/3 + 3*1/5
sage: A - B
2/3 - 3*1/5
sage: B*3
9*1/5
sage: 2*A
2*2/3
sage: list(2*A + A)
[(3, 2/3)]

Python

>>> from sage.all import *
>>> A = FormalSum([(Integer(1), Integer(2)/Integer(3))]); A
2/3
>>> B = FormalSum([(Integer(3), Integer(1)/Integer(5))]); B
3*1/5
>>> -B
-3*1/5
>>> A + B
2/3 + 3*1/5
>>> A - B
2/3 - 3*1/5
>>> B*Integer(3)
9*1/5
>>> Integer(2)*A
2*2/3
>>> list(Integer(2)*A + A)
[(3, 2/3)]

Sage Live

A = FormalSum([(1, 2/3)]); A
B = FormalSum([(3, 1/5)]); B
-B
A + B
A - B
B*3
2*A
list(2*A + A)

class sage.structure.formal_sum.FormalSum(x, parent=None, check=True, reduce=True)

Bases: ModuleElement

A formal sum over a ring.

reduce()

EXAMPLES:

Sage

sage: a = FormalSum([(-2,3), (2,3)], reduce=False); a
-2*3 + 2*3
sage: a.reduce()
sage: a
0

Python

>>> from sage.all import *
>>> a = FormalSum([(-Integer(2),Integer(3)), (Integer(2),Integer(3))], reduce=False); a
-2*3 + 2*3
>>> a.reduce()
>>> a
0

Sage Live

a = FormalSum([(-2,3), (2,3)], reduce=False); a
a.reduce()
a

class sage.structure.formal_sum.FormalSums

Bases: UniqueRepresentation, Module

The R-module of finite formal sums with coefficients in some ring R.

EXAMPLES:

Sage

sage: FormalSums()
Abelian Group of all Formal Finite Sums over Integer Ring
sage: FormalSums(ZZ)
Abelian Group of all Formal Finite Sums over Integer Ring
sage: FormalSums(GF(7))
Abelian Group of all Formal Finite Sums over Finite Field of size 7
sage: FormalSums(ZZ[sqrt(2)])                                                   # needs sage.rings.number_field sage.symbolic
Abelian Group of all Formal Finite Sums over
 Maximal Order generated by sqrt2 in Number Field in sqrt2
 with defining polynomial x^2 - 2 with sqrt2 = 1.414213562373095?
sage: FormalSums(GF(9,'a'))                                                     # needs sage.rings.finite_rings
Abelian Group of all Formal Finite Sums over Finite Field in a of size 3^2

Python

>>> from sage.all import *
>>> FormalSums()
Abelian Group of all Formal Finite Sums over Integer Ring
>>> FormalSums(ZZ)
Abelian Group of all Formal Finite Sums over Integer Ring
>>> FormalSums(GF(Integer(7)))
Abelian Group of all Formal Finite Sums over Finite Field of size 7
>>> FormalSums(ZZ[sqrt(Integer(2))])                                                   # needs sage.rings.number_field sage.symbolic
Abelian Group of all Formal Finite Sums over
 Maximal Order generated by sqrt2 in Number Field in sqrt2
 with defining polynomial x^2 - 2 with sqrt2 = 1.414213562373095?
>>> FormalSums(GF(Integer(9),'a'))                                                     # needs sage.rings.finite_rings
Abelian Group of all Formal Finite Sums over Finite Field in a of size 3^2

Sage Live

FormalSums()
FormalSums(ZZ)
FormalSums(GF(7))
FormalSums(ZZ[sqrt(2)])                                                   # needs sage.rings.number_field sage.symbolic
FormalSums(GF(9,'a'))                                                     # needs sage.rings.finite_rings

Element

alias of FormalSum

base_extend(R)

EXAMPLES:

Sage

sage: F7 = FormalSums(ZZ).base_extend(GF(7)); F7
Abelian Group of all Formal Finite Sums over Finite Field of size 7

Python

>>> from sage.all import *
>>> F7 = FormalSums(ZZ).base_extend(GF(Integer(7))); F7
Abelian Group of all Formal Finite Sums over Finite Field of size 7

Sage Live

F7 = FormalSums(ZZ).base_extend(GF(7)); F7

The following tests against a bug that was fixed at Issue #18795:

Sage

sage: isinstance(F7, F7.category().parent_class)
True

Python

>>> from sage.all import *
>>> isinstance(F7, F7.category().parent_class)
True

Sage Live

isinstance(F7, F7.category().parent_class)

Make live