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

TESTS:

sage: R = FormalSums(QQ)
sage: loads(dumps(R)) == R
True
sage: a = R(2/3) + R(-5/7); a
-5/7 + 2/3
sage: loads(dumps(a)) == a
True
class sage.structure.formal_sum.FormalSum(x, parent=Abelian Group of all Formal Finite Sums over Integer Ring, check=True, reduce=True)

Bases: sage.structure.element.ModuleElement

A formal sum over a ring.

reduce()

EXAMPLES:

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

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.modules.module.Module_old

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

EXAMPLES:

sage: FormalSums()
Abelian Group of all Formal Finite Sums over Integer Ring
sage: FormalSums(ZZ[sqrt(2)])
Abelian Group of all Formal Finite Sums over Order in Number Field in sqrt2 with defining polynomial x^2 - 2
sage: FormalSums(GF(9,'a'))
Abelian Group of all Formal Finite Sums over Finite Field in a of size 3^2
base_extend(R)

EXAMPLES:

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

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