return the content of an Ore polynomial - Maple Help

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OreTools[Content] - return the content of an Ore polynomial

OreTools[Primitive] - return primitive part of an Ore polynomial

OreTools[MonicAssociate] - return left or right monic associate of an Ore polynomial

OreTools[Normalize] - return the normal form of an Ore polynomial

Calling Sequence

Content(Poly, 'p')

Primitive(Poly, 'c')

MonicAssociate['left'](Poly, 's')

MonicAssociate(Poly, 's')

MonicAssociate['right'](Poly, A, 's')

Normalize(Poly)

Parameters

Poly

-

Ore polynomial; to define an Ore polynomial, use the OrePoly structure.

A

-

Ore algebra; to define an Ore algebra, use the SetOreRing function.

c, p, s

-

(optional) names

Description

• 

The Content(Poly, 'p') calling sequence returns the content of the Ore polynomial Poly. If the second (optional) argument p is present, the primitive part of Poly is assigned to p.

• 

The Primitive(Poly, 'c') calling sequence returns the primitive part of the Ore poly Poly. If the second (optional) argument c is present, the content of Poly is assigned to c.

• 

If the coefficients of Poly are integral (commutative) polynomials, then its content c is the gcd of its coefficients and its primitive part is equal to (1/c) Poly.

• 

If the coefficients of Poly are rational functions, then its content c and primitive part pp satisfy:

1. 

The primitive part pp is an Ore polynomial with integral (commutative) polynomial coefficients whose content is 1. Poly = c pp

• 

The MonicAssociate['left'](Poly, 's') or MonicAssociate(Poly, 's')  calling sequence returns (1/l) Poly where l is the leading coefficient of Poly. If the second (optional) argument s is present, (1/l) is assigned to l.

• 

The MonicAssociate['right'](Poly, A, 's') calling sequence returns Poly a,  where a belongs to the coefficient field such that the product (Poly a) is monic. If the third (optional) argument s, is present, a is assigned to s.

• 

The Normalize(Poly) calling sequence returns Poly with nonzero leading coefficient when Poly is nonzero; returns 'OrePoly'(0), otherwise.

Examples

withOreTools:

Ore1:=OrePolyn,n2,n3

Ore1:=OrePolyn,n2,n3

(1)

ContentOre1,'pp'

n

(2)

pp

OrePoly1,n,n2

(3)

PrimitiveOre1,'c'

OrePoly1,n,n2

(4)

c

n

(5)

MonicAssociate'left'Ore1,'s'

OrePoly1n2,1n,1

(6)

s

n3

(7)

MonicAssociate'OrePoly'0

OrePoly0

(8)

A:=SetOreRingn,'shift'

A:=UnivariateOreRingn,shift

(9)

MonicAssociate'right'Ore1,A,'s'

OrePolynn23,n2n13,1

(10)

s

n23

(11)

Normalize'OrePoly'0,n,n1,0,0

OrePoly0,n,n1

(12)

PrimitiveOre1,'c'

OrePoly1,n,n2

(13)

c

n

(14)

See Also

OreTools, OreTools/Arith, OreTools/OreAlgebra, OreTools/OrePoly


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