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.

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

sage.structure.formal_sum.FormalSums(R=Integer Ring)

Return the R-module of finite formal sums with coefficients in R.

INPUT:

R – a ring (default: ZZ)

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

class sage.structure.formal_sum.FormalSums_generic(base=Integer Ring)

Bases: sage.modules.module.Module

base_extend(R)

EXAMPLES:

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

get_action_impl(other, op, self_is_left)

EXAMPLES:

sage: A = FormalSums(RR).get_action(RR); A     # indirect doctest
Right scalar multiplication by Real Field with 53 bits of precision on Abelian Group of all Formal Finite Sums over Real Field with 53 bits of precision

sage: A = FormalSums(ZZ).get_action(QQ); A
Right scalar multiplication by Rational Field on Abelian Group of all Formal Finite Sums over Rational Field
with precomposition on left by Call morphism:
  From: Abelian Group of all Formal Finite Sums over Integer Ring
  To:   Abelian Group of all Formal Finite Sums over Rational Field
sage: A = FormalSums(QQ).get_action(ZZ); A
Right scalar multiplication by Integer Ring on Abelian Group of all Formal Finite Sums over Rational Field