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OreTools[Modular]

  

GCRD

  

compute the GCRD of two Ore polynomials modulo a prime

  

LCLM

  

compute the LCLM of a sequence of Ore polynomials modulo a prime

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

Modular[GCRD](Ore1, Ore2, p, A)

Modular[LCLM](Ore1, Ore2, ..., Orek, p, A)

Parameters

Ore1, Ore2, ... Orek

-

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

Description

• 

The Modular[GCRD](Ore1, Ore2, p, A) calling sequence returns the GCRD of Ore1 and Ore2 modulo the prime p.

• 

The Modular[LCLM](Ore1, Ore2, ..., Orek, p, A) calling sequence returns the GCRD of Ore1, Ore2, ..., Orek modulo the prime p.

Examples

with(OreTools):

A := SetOreRing(n, 'differential');

AUnivariateOreRingn,differential

(1)

Ore1 := 'OrePoly'(-n, n+6);

Ore1OrePolyn,n+6

(2)

Ore2 := 'OrePoly'(-n, n-1);

Ore2OrePolyn,n1

(3)

Ore3 := 'OrePoly'(n+1, n-1);

Ore3OrePolyn+1,n1

(4)

Poly1 := Modular[Multiply](Ore1, Ore3, 541, A);

Poly1OrePoly540n2+6,9n+12,n+6n+540

(5)

Poly2 := Modular[Multiply](Ore2, Ore3, 541, A);

Poly2OrePoly540n2+540,2n+539,n+5402

(6)

Modular[GCRD](Poly1, Poly2, 541, A);

OrePolyn+1n+540,1

(7)

Modular[LCLM](Poly1, Poly2, Ore1, 541, A);

OrePoly540n6+531n5+37n4+227n3+468n2+95n+456n6+24n5+187n4+511n3+177n2+181n+1,2n6+38n5+168n4+405n3+438n2+101n+195n6+24n5+187n4+511n3+177n2+181n+1,519n5+169n4+97n3+68n2+439n+267n6+24n5+187n4+511n3+177n2+181n+1,539n5+523n4+88n3+396n2+200n+510n5+18n4+79n3+37n2+496n+451,1

(8)

References

  

Abramov, S.A.; Le, H.Q.; and Li, Z. "OreTools: a computer algebra library for univariate Ore polynomial rings." Technical Report CS-2003-12. School of Computer Science, University of Waterloo, 2003.

  

Li, Z., and Nemes, I. "A modular algorithm for computing greatest common right divisors of Ore polynomials." Proc. of ISSAC'97, pp. 282-289. Edited by W. Kuechlin. ACM Press, 1997.

See Also

OreTools

OreTools/Euclidean

OreTools/Modular

OreTools/OreAlgebra

OreTools/OrePoly

OreTools/SetOreRing