The GCD['right'](Poly1, Poly2, ..., Polyk, A) or GCD(Poly1, Poly2, ..., Polyk, A) calling sequence returns the monic GCRD of Poly1, Poly2, ..., Polyk. The GCD['left'](Poly1, Poly2, ..., Polyk, A) calling sequence returns the monic GCLD of Poly1, Poly2, ..., Polyk.

If the optional 'cofactors = true' equation is specified, the GCD function returns a list in which the first element is the GCRD (resp. GCLD) of the Ore polynomials and the second element is a list of all the left (resp. right) cofactors of the Ore polynomials and the GCRD (resp. GCLD).

The LCM['left'](Poly1, Poly2, ..., Polyk, A) or LCM(Poly1, Poly2, ..., Polyk, A) calling sequence returns the monic LCLM of Poly1, Poly2, ..., Polyk. The LCM['right'](Poly1, Poly2, ..., Polyk, A) calling sequence returns the monic LCRM of Poly1, Poly2, ..., Polyk.

If the optional 'cofactors = true' equation is specified, the LCM function returns a list in which the first element is the LCLM (resp. LCRM) of the Ore polynomials and the second element is a list of all the left (resp. right) cofactors of the LCLM (resp. LCRM) and the Ore polynomials.

The ExtendedGCD['right'](Poly1, Poly2, A) or ExtendedGCD(Poly1, Poly2, A) calling sequence returns the monic GCRD of Poly1 and Poly2.

The fourth optional argument c1, when present, is assigned a pair [u, v] of Ore polynomials such that (u Poly1 - G) is right divisible by Poly2, where G is the monic GCRD of Poly1 and Poly2; and v Poly1 is the LCLM of Poly1 and Poly2.

The fourth and fifth optional arguments c1 and c2, when present, are assigned two pairs [u1, v1] and [u2, v2] of Ore polynomials, respectively. In this case,

where G is the monic GCRD of Poly1 and Poly2, v1 Poly1 is the LCLM of Poly1 and Poly2.

The ExtendedGCD['left'](Poly1, Poly2, A) calling sequence returns the monic GCLD of Poly1 and Poly2.

The fourth optional argument c1, when present, is assigned a pair [u, v] of Ore polynomials such that (Poly1 u - G) is left divisible by Poly2, where G is the monic GCLD of Poly1 and Poly2; and Poly1 v is the LCRM of Poly1 and Poly2.

The fourth and fifth optional arguments c1 and c2, when present, are assigned two pairs [u1, v1] and [u2, v2] of Ore polynomials, respectively. In this case,

where G is the monic GCLD of Poly1 and Poly2, Poly1 v1 is the LCRM of Poly1 and Poly2.

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

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

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

$G≔\mathrm{OrePoly}\left(1\,3x\right)$

$G≔\mathrm{OrePoly}\left(1\,3x\right)$

$\mathrm{Poly1}≔\mathrm{Multiply}\left(\mathrm{OrePoly}\left(\frac{1}{x}\,0\,\frac{x^2}{x+2}\right)\,G\,A\right)$

$\mathrm{Poly1}≔\mathrm{OrePoly}\left(\frac{1}{x}\,3\,\frac{7x^2}{x+2}\,\frac{3x^3}{x+2}\right)$

$\mathrm{Poly2}≔\mathrm{Multiply}\left(\mathrm{OrePoly}\left(x-1\,x\right)\,G\,A\right)$

$\mathrm{Poly2}≔\mathrm{OrePoly}\left(x-1\,3x^2+x\,3x^2\right)$

$\mathrm{GCD}\left['\mathrm{right}'\right]\left(\mathrm{Poly1}\,\mathrm{Poly2}\,A\right)$

$\mathrm{OrePoly}\left(\frac{1}{3x}\,1\right)$

$\mathrm{GCD}\left(\mathrm{Poly1}\,G\,\mathrm{Poly2}\,A\right)$

$\mathrm{OrePoly}\left(\frac{1}{3x}\,1\right)$

$\mathrm{GCD}\left(\mathrm{Poly1}\,\mathrm{Poly2}\,G\,'\mathrm{cofactors}=\mathrm{true}'\,A\right)$

$\left[\mathrm{OrePoly}\left(\frac{1}{3x}\,1\right)\,\left[\mathrm{OrePoly}\left(3\,\frac{6x^2}{x+2}\,\frac{3x^3}{x+2}\right)\,\mathrm{OrePoly}\left(3x^2\,3x^2\right)\,\mathrm{OrePoly}\left(3x\right)\right]\right]$

$\mathrm{LCM}\left(\mathrm{Poly1}\,\mathrm{Poly2}\,G\,A\right)$

$\mathrm{OrePoly}\left(\frac{x^5-3x^4-7x^3+15x^2-4}{3x^5\left(x^3-2x^2+x+2\right)}\,\frac{3x^5-5x^4-21x^3+33x^2+16x+4}{3x^4\left(x^3-2x^2+x+2\right)}\,\frac{7x^5-18x^4+7x^3+12x^2+40x+12}{3\left(x^3-2x^2+x+2\right)x^3}\,\frac{3x^4+x^3-17x^2+19x+32}{3x\left(x^3-2x^2+x+2\right)}\,1\right)$

$H≔\mathrm{ExtendedGCD}\left['\mathrm{right}'\right]\left(\mathrm{Poly1}\,\mathrm{Poly2}\,A\,'\mathrm{c1}'\right)$

$H≔\mathrm{OrePoly}\left(\frac{x^3-2x^2+x+2}{x\left(x+2\right)}\,\frac{3\left(x^3-2x^2+x+2\right)}{x+2}\right)$

$\mathrm{c1}$

$\left[\mathrm{OrePoly}\left(1\right)\,\mathrm{OrePoly}\left(-\frac{x^2\left(x^4-2x^3-7x^2+12x+4\right)}{{\left(x^3-2x^2+x+2\right)}^2}\,-\frac{x^2\left(x+2\right)}{x^3-2x^2+x+2}\right)\right]$

$\mathrm{Remainder}\left['\mathrm{right}'\right]\left(\mathrm{Minus}\left(\mathrm{Multiply}\left(\mathrm{c1}\left[1\right]\,\mathrm{Poly1}\,A\right)\,H\right)\,\mathrm{Poly2}\,A\right)$

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

$\mathrm{Remainder}\left(\mathrm{Multiply}\left(\mathrm{c1}\left[2\right]\,\mathrm{Poly1}\,A\right)\,\mathrm{Poly2}\,A\right)$

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

$H≔\mathrm{ExtendedGCD}\left(\mathrm{Poly1}\,\mathrm{Poly2}\,A\,'\mathrm{c1}'\,'\mathrm{c2}'\right)$

$H≔\mathrm{OrePoly}\left(\frac{x^3-2x^2+x+2}{x\left(x+2\right)}\,\frac{3\left(x^3-2x^2+x+2\right)}{x+2}\right)$

$\mathrm{Minus}\left(\mathrm{Add}\left(\mathrm{Multiply}\left(\mathrm{c1}\left[1\right]\,\mathrm{Poly1}\,A\right)\,\mathrm{Multiply}\left(\mathrm{c2}\left[1\right]\,\mathrm{Poly2}\,A\right)\right)\,H\right)$

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

$\mathrm{Add}\left(\mathrm{Multiply}\left(\mathrm{c1}\left[2\right]\,\mathrm{Poly1}\,A\right)\,\mathrm{Multiply}\left(\mathrm{c2}\left[2\right]\,\mathrm{Poly2}\,A\right)\right)$

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

$A≔\mathrm{SetOreRing}\left(n\,'\mathrm{shift}'\right)$

$A≔\mathrm{UnivariateOreRing}\left(n\,\mathrm{shift}\right)$

$G≔\mathrm{OrePoly}\left(1\,3n-3\right)$

$G≔\mathrm{OrePoly}\left(1\,3n-3\right)$

$\mathrm{Poly1}≔\mathrm{Multiply}\left(G\,\mathrm{OrePoly}\left(1\,n\,n^2\right)\,A\right)$

$\mathrm{Poly1}≔\mathrm{OrePoly}\left(1\,4n-3\,4n^2-3\,3\left(n-1\right){\left(n+1\right)}^2\right)$

$\mathrm{Poly2}≔\mathrm{Multiply}\left(G\,\mathrm{OrePoly}\left(n-1\,n\right)\,G\,A\right)$

$\mathrm{Poly2}≔\mathrm{OrePoly}\left(n-1\,6n^2-8n+3\,9n^3-3n^2-3\,9\left(n-1\right){\left(n+1\right)}^2\right)$

$\mathrm{GCD}\left['\mathrm{left}'\right]\left(\mathrm{Poly1}\,\mathrm{Poly2}\,A\right)$

$\mathrm{OrePoly}\left(\frac{1}{3\left(n-2\right)}\,1\right)$

$\mathrm{GCD}\left['\mathrm{left}'\right]\left(\mathrm{Poly1}\,\mathrm{Poly2}\,G\,'\mathrm{cofactors}=\mathrm{true}'\,A\right)$

$\left[\mathrm{OrePoly}\left(\frac{1}{3\left(n-2\right)}\,1\right)\,\left[\mathrm{OrePoly}\left(3n-6\,3n^2-6n\,3\left(n-2\right)n^2\right)\,\mathrm{OrePoly}\left(3n^2-9n+6\,9n^3-33n^2+39n-18\,9\left(n-2\right)n^2\right)\,\mathrm{OrePoly}\left(3n-6\right)\right]\right]$

$\mathrm{LCM}\left['\mathrm{right}'\right]\left(\mathrm{Poly1}\,\mathrm{Poly2}\,G\,A\right)$

$\mathrm{OrePoly}\left(\frac{7n^3-66n^2+176n-117}{9\left(7n^7-171n^6+1750n^5-9735n^4+31813n^3-61104n^2+63900n-28080\right)}\,\frac{49n^5-518n^4+1972n^3-3307n^2+2398n-648}{9\left(n^5-15n^4+87n^3-245n^2+336n-180\right)\left(7n^3-38n^2+58n-27\right)}\,\frac{2\left(56n^5-297n^4+430n^3-141n^2+48n-81\right)}{9\left(n^3-8n^2+20n-16\right)n\left(7n^2-17n+3\right)\left(n-1\right)}\,\frac{105n^5-122n^4-291n^3+260n^2+171n-144}{9\left(n^3-5n^2+7n-3\right)n\left(7n^2-3n-7\right)}\,\frac{21n^5+47n^4-8n^3-70n^2+4n+21}{3\left(n-2\right)\left(7n^2+11n-3\right)n^2}\,1\right)$

$H≔\mathrm{ExtendedGCD}\left['\mathrm{left}'\right]\left(\mathrm{Poly1}\,\mathrm{Poly2}\,A\,'\mathrm{c1}'\right)$

$H≔\mathrm{OrePoly}\left(\frac{1}{3\left(n-2\right)}\,1\right)$

$\mathrm{c1}$

$\left[\mathrm{OrePoly}\left(\frac{9n^4-96n^3+373n^2-633n+401}{3\left(7n^5-94n^4+492n^3-1259n^2+1580n-780\right)}\,\frac{3n^2-8n+3}{7n^3-45n^2+89n-54}\right)\,\mathrm{OrePoly}\left(\frac{\left(9n^5-111n^4+517n^3-1129n^2+1148n-434\right)\left(3n^2-14n+14\right)\left(3n^2-20n+31\right)}{27\left(7n^4-80n^3+332n^2-595n+390\right)\left(7n^5-94n^4+492n^3-1259n^2+1580n-780\right)}\,\frac{\left(21n^4-212n^3+762n^2-1165n+648\right){\left(3n^2-8n+3\right)}^2{\left(3n^2-14n+14\right)}^2}{27\left(7n^4-80n^3+332n^2-595n+390\right)\left(7n^2-31n+27\right)\left(7n^5-59n^4+186n^3-277n^2+197n-54\right)}\,\frac{{\left(3n^2-2n-2\right)}^2{\left(3n^2-8n+3\right)}^2}{9\left(7n^2-17n+3\right)n^2\left(7n^3-45n^2+89n-54\right)}\right)\right]$

$\mathrm{Remainder}\left['\mathrm{left}'\right]\left(\mathrm{Minus}\left(\mathrm{Multiply}\left(\mathrm{Poly1}\,\mathrm{c1}\left[1\right]\,A\right)\,H\right)\,\mathrm{Poly2}\,A\right)$

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

$\mathrm{Remainder}\left['\mathrm{left}'\right]\left(\mathrm{Multiply}\left(\mathrm{Poly1}\,\mathrm{c1}\left[2\right]\,A\right)\,\mathrm{Poly2}\,A\right)$

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

$H≔\mathrm{ExtendedGCD}\left['\mathrm{left}'\right]\left(\mathrm{Poly1}\,\mathrm{Poly2}\,A\,'\mathrm{c1}'\,'\mathrm{c2}'\right)$

$H≔\mathrm{OrePoly}\left(\frac{1}{3\left(n-2\right)}\,1\right)$

$\mathrm{Minus}\left(\mathrm{Add}\left(\mathrm{Multiply}\left(\mathrm{Poly1}\,\mathrm{c1}\left[1\right]\,A\right)\,\mathrm{Multiply}\left(\mathrm{Poly2}\,\mathrm{c2}\left[1\right]\,A\right)\right)\,H\right)$

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

$\mathrm{Add}\left(\mathrm{Multiply}\left(\mathrm{Poly1}\,\mathrm{c1}\left[2\right]\,A\right)\,\mathrm{Multiply}\left(\mathrm{Poly2}\,\mathrm{c2}\left[2\right]\,A\right)\right)$

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

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.