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.

$\mathrm{with}\left(\mathrm{OreTools}\right)\:$

$A≔\mathrm{SetOreRing}\left(x\,'\mathrm{differential}'\right)$

$A≔\mathrm{UnivariateOreRing}\left(x\,\mathrm{differential}\right)$

$\mathrm{Ore1}≔\mathrm{OrePoly}\left(1\,3x\,x^2-\left(1+x\right)\,23x+1\right)$

$\mathrm{Ore1}≔\mathrm{OrePoly}\left(1\,3x\,x^2-x-1\,23x+1\right)$

$\mathrm{Ore2}≔\mathrm{OrePoly}\left(x\,x\,x^2+x+1\right)$

$\mathrm{Ore2}≔\mathrm{OrePoly}\left(x\,x\,x^2+x+1\right)$

$U≔\mathrm{Modular}\left[\mathrm{RightEuclidean}\right]\left(\mathrm{Ore1}\,\mathrm{Ore2}\,11\,A\,'\mathrm{c1}'\,'\mathrm{c2}'\right)$

$U≔\left[4\,S\right]$

$m≔U\left[1\right]\;$$S≔U\left[2\right]$

$m≔4$

$S≔S$

$\mathrm{print}\left(S\right)$

$\left[\begin{array}{cccc}\mathrm{OrePoly}\left(1\,3x\,x^2+10x+10\,1+x\right)& \mathrm{OrePoly}\left(x\,x\,x^2+x+1\right)& \mathrm{OrePoly}\left(\frac{10x^5+x^4+5x^3+7x^2+2x}{x^4+2x^3+3x^2+2x+1}\,\frac{2x^5+5x^4+10x^3+8x^2+2x+10}{x^4+2x^3+3x^2+2x+1}\right)& \mathrm{OrePoly}\left(\frac{3x^{12}+4x^{11}+5x^{10}+x^9+8x^8+3x^7+8x^6+3x^5+6x^4+3x^3+6x^2+7x+6}{x^{10}+5x^9+8x^8+3x^6+7x^4+3x^3+8x^2+10x+3}\right)\end{array}\right]$

$$\begin{gathered} \mathbf{for}i\mathbf{to}m\mathbf{do} \\ \mathrm{W1}≔\mathrm{Modular}\left[\mathrm{Multiply}\right]\left(\mathrm{c1}\left[i\right]\,\mathrm{Ore1}\,11\,A\right); \\ \mathrm{W2}≔\mathrm{Modular}\left[\mathrm{Multiply}\right]\left(\mathrm{c2}\left[i\right]\,\mathrm{Ore2}\,11\,A\right); \\ W≔\mathrm{Modular}\left[\mathrm{Add}\right]\left(\mathrm{W1}\,\mathrm{W2}\,11\right); \\ C≔\mathrm{Modular}\left[\mathrm{Minus}\right]\left(W\,S\left[i\right]\,11\right); \\ \mathrm{print}\left(C\right) \\ \mathbf{end}\mathbf{do}\: \end{gathered}$$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{W3}≔\mathrm{Modular}\left[\mathrm{Multiply}\right]\left(\mathrm{c1}\left[5\right]\,\mathrm{Ore1}\,11\,A\right)\:$

$\mathrm{W4}≔\mathrm{Modular}\left[\mathrm{Multiply}\right]\left(\mathrm{c2}\left[5\right]\,\mathrm{Ore2}\,11\,A\right)\:$

$\mathrm{Modular}\left[\mathrm{Add}\right]\left(\mathrm{W3}\,\mathrm{W4}\,11\right)$

$\mathrm{OrePoly}\left(0\right)$

$A≔\mathrm{SetOreRing}\left(x\,'\mathrm{differential}'\right)$

$A≔\mathrm{UnivariateOreRing}\left(x\,\mathrm{differential}\right)$

$\mathrm{Ore1}≔\mathrm{OrePoly}\left(1\,3x\,x^2-\left(1+x\right)\,23x+1\right)$

$\mathrm{Ore1}≔\mathrm{OrePoly}\left(1\,3x\,x^2-x-1\,23x+1\right)$

$\mathrm{Ore2}≔\mathrm{OrePoly}\left(x\,x\,x^2+x+1\right)$

$\mathrm{Ore2}≔\mathrm{OrePoly}\left(x\,x\,x^2+x+1\right)$

$U≔\mathrm{Modular}\left[\mathrm{FractionFreeRightEuclidean}\right]\left(\mathrm{Ore1}\,\mathrm{Ore2}\,11\,A\,'\mathrm{c1}'\,'\mathrm{c2}'\right)$

$U≔\left[4\,S\right]$

$m≔U\left[1\right]\;$$S≔U\left[2\right]$

$m≔4$

$S≔S$

$\mathrm{print}\left(S\right)$

$\left[\begin{array}{cccc}\mathrm{OrePoly}\left(1\,3x\,x^2+10x+10\,1+x\right)& \mathrm{OrePoly}\left(x\,x\,x^2+x+1\right)& \mathrm{OrePoly}\left(10x^5+x^4+5x^3+7x^2+2x\,2x^5+5x^4+10x^3+8x^2+2x+10\right)& \mathrm{OrePoly}\left(x^8+3x^7+4x^5+6x^4+7x^3+3x^2+2x+2\right)\end{array}\right]$

$$\begin{gathered} \mathbf{for}i\mathbf{to}m\mathbf{do} \\ \mathrm{W1}≔\mathrm{Modular}\left[\mathrm{Multiply}\right]\left(\mathrm{c1}\left[i\right]\,\mathrm{Ore1}\,11\,A\right); \\ \mathrm{W2}≔\mathrm{Modular}\left[\mathrm{Multiply}\right]\left(\mathrm{c2}\left[i\right]\,\mathrm{Ore2}\,11\,A\right); \\ W≔\mathrm{Modular}\left[\mathrm{Add}\right]\left(\mathrm{W1}\,\mathrm{W2}\,11\right); \\ C≔\mathrm{Modular}\left[\mathrm{Minus}\right]\left(W\,S\left[i\right]\,11\right); \\ \mathrm{print}\left(C\right) \\ \mathbf{end}\mathbf{do}\: \end{gathered}$$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{W3}≔\mathrm{Modular}\left[\mathrm{Multiply}\right]\left(\mathrm{c1}\left[5\right]\,\mathrm{Ore1}\,11\,A\right)\:$

$\mathrm{W4}≔\mathrm{Modular}\left[\mathrm{Multiply}\right]\left(\mathrm{c2}\left[5\right]\,\mathrm{Ore2}\,11\,A\right)\:$

$\mathrm{Modular}\left[\mathrm{Add}\right]\left(\mathrm{W3}\,\mathrm{W4}\,11\right)$

$\mathrm{OrePoly}\left(0\right)$

Li, Z. "A subresultant theory for Ore polynomials with applications." Proc. of ISSAC'98, pp.132-139. Edited by O. Gloor. ACM Press, 1998.