perform a fraction-free version of right Euclidean algorithm (usual, half-extended, and extended) modulo a prime

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OreTools[Modular][FractionFreeRightEucliean] - perform a fraction-free version of right Euclidean algorithm (usual, half-extended, and extended) modulo a prime

OreTools[Modular][RightEuclidean] - perform right Euclidean algorithm (usual, half-extended, and extended)

	Calling Sequence
	Modular[FractionFreeRightEuclidean](Poly1, Poly2, p, A, 'c1', 'c2') Modular[RightEuclidean](Poly1, Poly2, p, A, 'c1', 'c2')

Parameters

Poly1, Poly2	-	nonzero Ore polynomials; to define an Ore polynomial, use the OrePoly structure
p	-	prime
A	-	Ore algebra; to define an Ore algebra, use the SetOreRing command
'c1', 'c2'	-	(optional) unevaluated names

Description

•	Modular[FractionFreeRightEuclidean](Poly1, Poly2, p, A, 'c1', 'c2') calling sequence returns a list [m, S] where m is a positive integer and S is an array with m elements storing the subresultant sequence of the first kind of Poly1 and Poly2.

The Modular[FractionFreeRightEuclidean] command requires that Poly1 and Poly2 be fraction-free, and that the commutation rule of the Ore algebra A also be fraction-free.

•	If the optional fourth argument to the FractionFreeRightEuclidean command c1 is specified, it is assigned the first co-sequence of Poly1 and Poly2 so that:

and c1[m+1] Poly2 is a least common left multiple (LCLM) of Poly1 and Poly2.

•	If the optional fifth argument to the FractionFreeRightEuclidean command c2 is specified, it is assigned the second co-sequence of Poly1 and Poly2 so that:

and c1[m+1] Poly2 = - c2[m+1] Poly1 mod p is an LCLM of Poly1 and Poly2.

•	Modular[RightEuclidean](Poly1, Poly2, p, A, 'c1', 'c2') calling sequence returns a list [m, S] where m is a positive integer and S is an array with m elements storing the right Euclidean polynomial remainder sequence of Poly1 and Poly2.

•	If the optional fourth argument to the FractionFreeRightEuclidean command c1 is specified, it is assigned the first co-sequence of Poly1 and Poly2 so that:

and c1[m+1] Poly2 is a least common left multiple (LCLM) of Poly1 and Poly2.

•	If the optional fifth argument to the Modular[RightEuclidean] command c2 is specified, it is assigned the second co-sequence of Poly1 and Poly2 so that:

and c1[m+1] Poly2 = - c2[m+1] Poly1 mod p is an LCLM of Poly1 and Poly2.

Examples

(1)

(2)

(3)

(4)

(5)

vector([OrePoly(1, 3*x, x^2+10+10*x, 1+x), OrePoly(x, x, x^2+x+1), OrePoly((2*x+7*x^2+5*x^3+x^4+10*x^5)/(1+2*x+3*x^2+2*x^3+x^4), (2*x+8*x^2+10*x^3+5*x^4+2*x^5+10)/(1+2*x+3*x^2+2*x^3+x^4)), OrePoly((7*x+6*x^2+3*x^3+6*x^4+3*x^5+3*x^7+x^9+5*x^10+4*x^11+6+8*x^6+8*x^8+3*x^12)/(3+10*x+8*x^2+3*x^3+7*x^4+3*x^6+8*x^8+5*x^9+x^10))])

(6)

Check the co-sequences.

for i to m do W1 := Modular[Multiply](c1[i], Ore1, 11, A); W2 := Modular[Multiply](c2[i], Ore2, 11, A); W := Modular[Add](W1, W2, 11); C := Modular[Minus](W, S[i], 11); print(C) end do

(7)

Check the LCLM.

(8)

Try fraction-free right Euclidean algorithm.

(9)

(10)

(11)

(12)

(13)

(14)

Check the co-sequences.

for i to m do W1 := Modular[Multiply](c1[i], Ore1, 11, A); W2 := Modular[Multiply](c2[i], Ore2, 11, A); W := Modular[Add](W1, W2, 11); C := Modular[Minus](W, S[i], 11); print(C) end do