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OreTools[Modular]

  

RightPseudoQuotient

  

compute the right pseudo-quotient of two Ore polynomials modulo a prime

  

RightPseudoRemainder

  

compute the right pseudo-remainder of two Ore polynomials modulo a prime

  

RightQuotient

  

compute the right quotient of two Ore polynomials modulo a prime

  

RightRemainder

  

compute the right remainder of two Ore polynomials modulo a prime

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Modular[RightPseudoQuotient](Ore1, Ore2, p, A, 'm', 'R')

Modular[RightPseudoRemainder](Ore1, Ore2, p, 'm', 'Q')

Modular[RightQuotient](Ore1, Ore2, p, A, 'R')

Modular[RightRemainder](Ore1, Ore2, p, A, 'Q')

Parameters

Ore1, Ore2

-

two Ore polynomials; to define an Ore polynomial, use the OrePoly structure

p

-

prime

A

-

Ore ring; to define an Ore ring, use the SetOreRing command

m, Q, R

-

unevaluated names

Description

• 

The Modular[RightPseudoQuotient](Ore1, Ore2, p, A, 'm', 'R') calling sequence computes the right pseudo-quotient of Ore1 and Ore2 modulo the prime p. If the fifth (optional) argument is present, it is assigned the multiplier. If the sixth (optional) argument is present, it is assigned the right pseudo-remainder of  Ore1  and Ore2.

• 

The Modular[RightPseudoRemainder](Ore1, Ore2, p, A, 'm', 'Q') calling sequence computes the right pseudo-remainder of Ore1 and Ore2 modulo the prime p. If the fifth (optional) argument is present, it is assigned the multiplier. If the sixth (optional) argument is present, it is assigned the right pseudo-quotient of Ore1  and Ore2.

• 

The Modular[RightQuotient](Ore1, Ore2, p, A, ''R') calling sequence computes the right quotient of Ore1 and Ore2 modulo the prime p. If the fifth (optional) argument is present, it is assigned the right remainder of  Ore1  and Ore2.

• 

The Modular[RightRemainder](Ore1, Ore2, p, A, 'Q') calling sequence computes the right remainder of Ore1 and Ore2 modulo the prime p. If the fifth (optional) argument is present, it is assigned the right quotient of Ore1  and Ore2.

Examples

withOreTools:

ASetOreRingn,differential

A:=UnivariateOreRingn,differential

(1)

Ore1'OrePoly'n,5n+n2+3,n3,38n21

Ore1:=OrePolyn,n2+5n3,n3,38n21

(2)

Ore2'OrePoly'n,5n+n2+3,n3

Ore2:=OrePolyn,n2+5n3,n3

(3)

Modular[RightPseudoRemainder]Ore1,Ore2,11,A

OrePoly5n5+8n4+3n3+n2+8n+3,5n6+5n5+10n4+2n3+6n2+6n+1

(4)

QModular[RightPseudoQuotient]Ore1,Ore2,19,A,'m','R'

Q:=OrePoly18n+7,18n+3

(5)

Check the results.

lModular[ScalarMultiply]m,Ore1,19

l:=OrePoly18n2+13n+9n,18n2+13n+9n2+14n+3,n2+13n+9n+16,18n2+6n+10

(6)

rModular[Multiply]Q,Ore2,19,A

r:=OrePolyn2+13n+16,n3+10n2+5n+13,n3+10n2+8n+11,18n+162

(7)

Modular[Minus]Modular[Minus]l,r,19,R,19

OrePoly0

(8)

RModular[RightRemainder]Ore1,Ore2,11,A,'Q'

R:=OrePoly5n4+n3+6n2+8n+10n+8,5n5+9n4+4n3+3n2+4n+7n+8

(9)

Check the results.

rModular[Multiply]Q,Ore2,11,A

r:=OrePoly6n4+10n3+4n2+6n+1n+8,6n5+2n4+6n3+5n2+2n+8,n+8,5n+3n+8

(10)

Modular[Minus]Modular[Minus]Ore1,r,11,R,11

OrePoly0

(11)

Modular[RightQuotient]Ore1,Ore2,19,A

OrePoly18n+7n2+13n+9,18n+16

(12)

See Also

OreTools

OreTools/Euclidean

OreTools/Modular

OreTools/OreAlgebra

OreTools/OrePoly

OreTools/RingArith

OreTools[SetOreRing]

 


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