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Ore_algebra

 annihilators
 skew lcm of a pair of operators
 skew_gcdex
 extended skew gcd computation
 skew_pdiv
 skew pseudo-division
 skew_prem
 skew pseudo-remainder
 skew_elim
 skew elimination of an indeterminate

 Calling Sequence annihilators(p, q, A) skew_gcdex(p, q, x, A, opt) skew_pdiv(p, q, x, A) skew_prem(p, q, x, A) skew_elim(p, q, x, A)

Parameters

 p, q - skew polynomials A - Ore algebra table x - indeterminate of the algebra opt - (optional) literal string; one of $"monic"$, $"left"$, and $"left_monic"$

Description

 • The annihilators, skew_gcdex, skew_pdiv, skew_prem, and skew_elim commands perform simple algebraic operations in Ore algebras, all based on skew pseudo-division and skew Euclidean algorithms.
 • The skew_pdiv(p, q, x, A) function performs a skew pseudo-division of the skew polynomial p by the skew polynomial q.  Both polynomials are viewed as polynomials in x in the Ore algebra A.  The function returns a list $\left[u,v,r\right]$ such that $up-vq=r$ is of degree lower than q.  The resulting v is a polynomial in x, whereas u is a coefficient.  The skew_prem(p, q, x, A) function simply returns the remainder r.
 • The skew_gcdex(p, q, x, A) function performs an extended skew gcd algorithm on the skew polynomials p and q viewed as polynomials in x with coefficients in their other indeterminates.  With no option or the option $"monic"$, it returns a list $\left[g,a,b,u,v\right]$ such that $\mathrm{up}+\mathrm{vq}=0$ and $\mathrm{ap}+\mathrm{bq}=g$.  Hence, g is a right gcd of p and q (in an algebra where all coefficient indeterminates are invertible), while up and vq are left lcms of p and q.  Without the option, g, a, and b are fraction-free polynomials with no common content, and u and v are fraction-free polynomials with no common (left) content; when the option $"monic"$ is used, the polynomial g is made monic and a and b are changed accordingly.  With the option "left" or "left_monic", skew_gcdex returns a list $\left[g,a,b,u,v\right]$ such that $\mathrm{pu}+\mathrm{qv}=0$ and $\mathrm{pa}+\mathrm{qb}=g$.  In this case, g is a left gcd.  The option "left" returns fraction-free polynomial while the option "left_monic" ensures that g is made monic (by multiplication by a fraction on the right).  (See also Ore_algebra[dual_algebra].)
 • The annihilators(p, q, A) function performs a specialized algorithm to return a list $\left[u,v\right]$ of skew polynomials of the algebra A such that $\mathrm{up}+\mathrm{vq}=0$.
 • The skew_elim(p, q, x, A) function tries to eliminate the indeterminate x between the skew polynomials p and q.  It returns a nonzero polynomial $\mathrm{ap}+\mathrm{bq}$ free from x, is such a polynomial exists.  Otherwise, a nonzero polynomial $\mathrm{ap}+\mathrm{bq}$ of least possible degree in x is returned.
 • The skew_gcdex, skew_pdiv, skew_prem, and skew_elim commands are specific to the case of skew polynomials viewed as polynomials in a single indeterminate.  A general (multivariate) treatment is provided via Groebner bases computations (see Groebner, and in particular Groebner[Basis], and Groebner[Reduce]).  However, skew_elim is appropriate to eliminate a single indeterminate between two skew polynomials without computing unneeded information.
 • These functions are part of the Ore_algebra package, and so can be used in the form annihilators(..), skew_gcdex(..), skew_pdiv(..), skew_prem(..), or skew_elim(..) only after performing the command with(Ore_algebra) or with(Ore_algebra,).  The functions can always be accessed in the long form Ore_algebra[annihilators](..), Ore_algebra[skew_gcdex](..), Ore_algebra[skew_pdiv], Ore_algebra[skew_prem], and Ore_algebra[skew_elim](..).

Examples

 > $\mathrm{with}\left(\mathrm{Ore_algebra}\right):$

Differential case.

 > $A≔\mathrm{diff_algebra}\left(\left[\mathrm{Dx},x\right]\right):$
 > $P≔\mathrm{skew_product}\left(x\left({\mathrm{Dx}}^{2}+\mathrm{Dx}-1\right),\left(xx-1\right)\left(\mathrm{Dx}\mathrm{Dx}+1\right),A\right):$
 > $Q≔\mathrm{skew_product}\left(\left(x-1\right)\left({\mathrm{Dx}}^{2}-\mathrm{Dx}+1\right),\left(xx-1\right)\left(\mathrm{Dx}\mathrm{Dx}+1\right),A\right):$

The skew polynomials can be viewed as polynomials in Dx

 > $\mathrm{skew_pdiv}\left(P,Q,\mathrm{Dx},A\right)$
 $\left[{x}{-}{1}{,}{x}{,}{2}{}{{\mathrm{Dx}}}^{{3}}{}{{x}}^{{4}}{-}{2}{}{{\mathrm{Dx}}}^{{3}}{}{{x}}^{{3}}{-}{2}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{4}}{-}{2}{}{{\mathrm{Dx}}}^{{3}}{}{{x}}^{{2}}{+}{6}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{3}}{+}{2}{}{\mathrm{Dx}}{}{{x}}^{{4}}{+}{2}{}{{\mathrm{Dx}}}^{{3}}{}{x}{-}{2}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{2}}{-}{2}{}{\mathrm{Dx}}{}{{x}}^{{3}}{-}{2}{}{{x}}^{{4}}{-}{2}{}{{\mathrm{Dx}}}^{{2}}{}{x}{-}{2}{}{\mathrm{Dx}}{}{{x}}^{{2}}{+}{6}{}{{x}}^{{3}}{+}{2}{}{\mathrm{Dx}}{}{x}{-}{2}{}{{x}}^{{2}}{-}{2}{}{x}\right]$ (1)
 > $G≔\mathrm{skew_gcdex}\left(P,Q,\mathrm{Dx},A\right)$
 ${G}{≔}\left[{2}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{6}}{-}{4}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{5}}{+}{2}{}{{x}}^{{6}}{+}{4}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{3}}{-}{4}{}{{x}}^{{5}}{-}{2}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{2}}{+}{4}{}{{x}}^{{3}}{-}{2}{}{{x}}^{{2}}{,}{1}{-}\left({{x}}^{{3}}{-}{2}{}{{x}}^{{2}}{+}{x}\right){}{\mathrm{Dx}}{+}{{x}}^{{2}}{-}{2}{}{x}{,}{2}{}{{x}}^{{3}}{-}{3}{}{{x}}^{{2}}{-}\left({-}{{x}}^{{3}}{+}{{x}}^{{2}}\right){}{\mathrm{Dx}}{,}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{5}}{-}{3}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{4}}{-}{\mathrm{Dx}}{}{{x}}^{{5}}{+}{3}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{3}}{+}{\mathrm{Dx}}{}{{x}}^{{4}}{+}{{x}}^{{5}}{-}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{2}}{+}{3}{}{\mathrm{Dx}}{}{{x}}^{{3}}{-}{2}{}{{x}}^{{4}}{-}{5}{}{\mathrm{Dx}}{}{{x}}^{{2}}{+}{2}{}{{x}}^{{3}}{+}{2}{}{\mathrm{Dx}}{}{x}{-}{4}{}{{x}}^{{2}}{+}{5}{}{x}{-}{2}{,}{-}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{5}}{+}{2}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{4}}{-}{\mathrm{Dx}}{}{{x}}^{{5}}{-}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{3}}{+}{4}{}{\mathrm{Dx}}{}{{x}}^{{4}}{+}{{x}}^{{5}}{-}{3}{}{\mathrm{Dx}}{}{{x}}^{{3}}{-}{{x}}^{{4}}{-}{2}{}{{x}}^{{3}}\right]$ (2)
 > $\mathrm{skew_product}\left(G\left[2\right],P,A\right)+\mathrm{skew_product}\left(G\left[3\right],Q,A\right)-G\left[1\right]$
 ${-}{4}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{3}}{+}{2}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{2}}{-}{2}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{6}}{+}{4}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{5}}{+}\left({4}{}{{x}}^{{5}}{-}{2}{}{{x}}^{{4}}{-}{8}{}{{x}}^{{3}}{+}{6}{}{{x}}^{{2}}\right){}{{\mathrm{Dx}}}^{{3}}{+}\left({-}{{x}}^{{6}}{+}{2}{}{{x}}^{{5}}{-}{2}{}{{x}}^{{3}}{+}{{x}}^{{2}}\right){}{{\mathrm{Dx}}}^{{5}}{+}\left({3}{}{{x}}^{{6}}{-}{2}{}{{x}}^{{5}}{-}{6}{}{{x}}^{{4}}{+}{6}{}{{x}}^{{3}}{-}{{x}}^{{2}}\right){}{{\mathrm{Dx}}}^{{2}}{+}\left({{x}}^{{6}}{-}{6}{}{{x}}^{{5}}{+}{2}{}{{x}}^{{4}}{+}{10}{}{{x}}^{{3}}{-}{7}{}{{x}}^{{2}}\right){}{\mathrm{Dx}}{+}\left({{x}}^{{6}}{+}{4}{}{{x}}^{{5}}{-}{12}{}{{x}}^{{4}}{+}{8}{}{{x}}^{{3}}{-}{{x}}^{{2}}\right){}{{\mathrm{Dx}}}^{{4}}{+}\left({-}{{x}}^{{6}}{+}{6}{}{{x}}^{{5}}{-}{2}{}{{x}}^{{4}}{-}{10}{}{{x}}^{{3}}{+}{7}{}{{x}}^{{2}}\right){}{\mathrm{Dx}}{+}\left({-}{4}{}{{x}}^{{5}}{+}{2}{}{{x}}^{{4}}{+}{8}{}{{x}}^{{3}}{-}{6}{}{{x}}^{{2}}\right){}{{\mathrm{Dx}}}^{{3}}{+}\left({-}{{x}}^{{6}}{-}{2}{}{{x}}^{{5}}{+}{6}{}{{x}}^{{4}}{-}{2}{}{{x}}^{{3}}{-}{{x}}^{{2}}\right){}{{\mathrm{Dx}}}^{{2}}{+}\left({{x}}^{{6}}{-}{2}{}{{x}}^{{5}}{+}{2}{}{{x}}^{{3}}{-}{{x}}^{{2}}\right){}{{\mathrm{Dx}}}^{{5}}{+}\left({-}{{x}}^{{6}}{-}{4}{}{{x}}^{{5}}{+}{12}{}{{x}}^{{4}}{-}{8}{}{{x}}^{{3}}{+}{{x}}^{{2}}\right){}{{\mathrm{Dx}}}^{{4}}$ (3)
 > $\mathrm{skew_product}\left(G\left[4\right],P,A\right)+\mathrm{skew_product}\left(G\left[5\right],Q,A\right)$
 $\left({{x}}^{{8}}{-}{3}{}{{x}}^{{7}}{+}{2}{}{{x}}^{{6}}{+}{2}{}{{x}}^{{5}}{-}{3}{}{{x}}^{{4}}{+}{{x}}^{{3}}\right){}{{\mathrm{Dx}}}^{{6}}{+}\left({8}{}{{x}}^{{7}}{-}{24}{}{{x}}^{{6}}{+}{24}{}{{x}}^{{5}}{-}{8}{}{{x}}^{{4}}\right){}{{\mathrm{Dx}}}^{{5}}{+}\left({12}{}{{x}}^{{6}}{-}{36}{}{{x}}^{{5}}{+}{36}{}{{x}}^{{4}}{-}{12}{}{{x}}^{{3}}\right){}{{\mathrm{Dx}}}^{{4}}{+}\left({2}{}{{x}}^{{8}}{-}{2}{}{{x}}^{{7}}{-}{8}{}{{x}}^{{6}}{+}{16}{}{{x}}^{{5}}{-}{10}{}{{x}}^{{4}}{+}{2}{}{{x}}^{{3}}\right){}{{\mathrm{Dx}}}^{{3}}{+}\left({-}{2}{}{{x}}^{{8}}{+}{10}{}{{x}}^{{7}}{-}{6}{}{{x}}^{{6}}{-}{22}{}{{x}}^{{5}}{+}{32}{}{{x}}^{{4}}{-}{12}{}{{x}}^{{3}}\right){}{{\mathrm{Dx}}}^{{2}}{+}\left({2}{}{{x}}^{{8}}{-}{10}{}{{x}}^{{7}}{+}{16}{}{{x}}^{{6}}{-}{8}{}{{x}}^{{5}}{-}{2}{}{{x}}^{{4}}{+}{2}{}{{x}}^{{3}}\right){}{\mathrm{Dx}}{+}\left({-}{{x}}^{{8}}{+}{3}{}{{x}}^{{7}}{-}{2}{}{{x}}^{{6}}{-}{2}{}{{x}}^{{5}}{+}{3}{}{{x}}^{{4}}{-}{{x}}^{{3}}\right){}{{\mathrm{Dx}}}^{{6}}{+}\left({-}{8}{}{{x}}^{{7}}{+}{24}{}{{x}}^{{6}}{-}{24}{}{{x}}^{{5}}{+}{8}{}{{x}}^{{4}}\right){}{{\mathrm{Dx}}}^{{5}}{+}\left({-}{12}{}{{x}}^{{6}}{+}{36}{}{{x}}^{{5}}{-}{36}{}{{x}}^{{4}}{+}{12}{}{{x}}^{{3}}\right){}{{\mathrm{Dx}}}^{{4}}{+}\left({-}{2}{}{{x}}^{{8}}{+}{2}{}{{x}}^{{7}}{+}{8}{}{{x}}^{{6}}{-}{16}{}{{x}}^{{5}}{+}{10}{}{{x}}^{{4}}{-}{2}{}{{x}}^{{3}}\right){}{{\mathrm{Dx}}}^{{3}}{+}\left({2}{}{{x}}^{{8}}{-}{10}{}{{x}}^{{7}}{+}{6}{}{{x}}^{{6}}{+}{22}{}{{x}}^{{5}}{-}{32}{}{{x}}^{{4}}{+}{12}{}{{x}}^{{3}}\right){}{{\mathrm{Dx}}}^{{2}}{+}\left({-}{2}{}{{x}}^{{8}}{+}{10}{}{{x}}^{{7}}{-}{16}{}{{x}}^{{6}}{+}{8}{}{{x}}^{{5}}{+}{2}{}{{x}}^{{4}}{-}{2}{}{{x}}^{{3}}\right){}{\mathrm{Dx}}$ (4)
 > $\mathrm{annihilators}\left(P,Q,A\right)$
 $\left[{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{5}}{-}{3}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{4}}{-}{\mathrm{Dx}}{}{{x}}^{{5}}{+}{3}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{3}}{+}{\mathrm{Dx}}{}{{x}}^{{4}}{+}{{x}}^{{5}}{-}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{2}}{+}{3}{}{\mathrm{Dx}}{}{{x}}^{{3}}{-}{2}{}{{x}}^{{4}}{-}{5}{}{\mathrm{Dx}}{}{{x}}^{{2}}{+}{2}{}{{x}}^{{3}}{+}{2}{}{\mathrm{Dx}}{}{x}{-}{4}{}{{x}}^{{2}}{+}{5}{}{x}{-}{2}{,}{-}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{5}}{+}{2}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{4}}{-}{\mathrm{Dx}}{}{{x}}^{{5}}{-}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{3}}{+}{4}{}{\mathrm{Dx}}{}{{x}}^{{4}}{+}{{x}}^{{5}}{-}{3}{}{\mathrm{Dx}}{}{{x}}^{{3}}{-}{{x}}^{{4}}{-}{2}{}{{x}}^{{3}}\right]$ (5)

or in x.  In this case, the algebra must be redefined accordingly.

 > $A≔\mathrm{diff_algebra}\left(\left[\mathrm{Dx},x\right],\mathrm{polynom}=x\right):$
 > $G≔\mathrm{skew_gcdex}\left(P,Q,x,A\right)$
 ${G}{≔}\left[{{\mathrm{Dx}}}^{{6}}{}{{x}}^{{2}}{-}{{\mathrm{Dx}}}^{{6}}{+}{8}{}{{\mathrm{Dx}}}^{{5}}{}{x}{+}{2}{}{{\mathrm{Dx}}}^{{4}}{}{{x}}^{{2}}{-}{2}{}{{\mathrm{Dx}}}^{{3}}{}{{x}}^{{2}}{+}{10}{}{{\mathrm{Dx}}}^{{4}}{+}{12}{}{{\mathrm{Dx}}}^{{3}}{}{x}{+}{2}{}{{\mathrm{Dx}}}^{{3}}{-}{4}{}{{\mathrm{Dx}}}^{{2}}{}{x}{-}{2}{}{\mathrm{Dx}}{}{{x}}^{{2}}{+}{14}{}{{\mathrm{Dx}}}^{{2}}{+}{4}{}{\mathrm{Dx}}{}{x}{-}{{x}}^{{2}}{+}{2}{}{\mathrm{Dx}}{-}{4}{}{x}{+}{3}{,}{{\mathrm{Dx}}}^{{2}}{-}{\mathrm{Dx}}{+}{1}{,}{-}{{\mathrm{Dx}}}^{{2}}{-}{\mathrm{Dx}}{+}{1}{,}{-}{{\mathrm{Dx}}}^{{8}}{}{x}{+}{{\mathrm{Dx}}}^{{8}}{+}{2}{}{{\mathrm{Dx}}}^{{7}}{}{x}{-}{10}{}{{\mathrm{Dx}}}^{{7}}{-}{4}{}{{\mathrm{Dx}}}^{{6}}{}{x}{+}{20}{}{{\mathrm{Dx}}}^{{6}}{+}{6}{}{{\mathrm{Dx}}}^{{5}}{}{x}{-}{34}{}{{\mathrm{Dx}}}^{{5}}{-}{7}{}{{\mathrm{Dx}}}^{{4}}{}{x}{+}{35}{}{{\mathrm{Dx}}}^{{4}}{+}{6}{}{{\mathrm{Dx}}}^{{3}}{}{x}{-}{34}{}{{\mathrm{Dx}}}^{{3}}{-}{2}{}{{\mathrm{Dx}}}^{{2}}{}{x}{+}{22}{}{{\mathrm{Dx}}}^{{2}}{-}{12}{}{\mathrm{Dx}}{+}{x}{+}{3}{,}{{\mathrm{Dx}}}^{{8}}{}{x}{+}{8}{}{{\mathrm{Dx}}}^{{7}}{+}{2}{}{{\mathrm{Dx}}}^{{6}}{-}{8}{}{{\mathrm{Dx}}}^{{5}}{-}{3}{}{{\mathrm{Dx}}}^{{4}}{}{x}{+}{14}{}{{\mathrm{Dx}}}^{{4}}{+}{4}{}{{\mathrm{Dx}}}^{{3}}{}{x}{-}{20}{}{{\mathrm{Dx}}}^{{3}}{-}{4}{}{{\mathrm{Dx}}}^{{2}}{}{x}{+}{18}{}{{\mathrm{Dx}}}^{{2}}{-}{8}{}{\mathrm{Dx}}{+}{x}{+}{4}\right]$ (6)
 > $\mathrm{skew_product}\left(G\left[2\right],P,A\right)+\mathrm{skew_product}\left(G\left[3\right],Q,A\right)-G\left[1\right]$
 ${0}$ (7)
 > $\mathrm{skew_product}\left(G\left[4\right],P,A\right)+\mathrm{skew_product}\left(G\left[5\right],Q,A\right)$
 ${0}$ (8)

Case of 'q'-calculus:

 > $A≔\mathrm{skew_algebra}\left(\mathrm{comm}=q,\mathrm{qdilat}=\left[\mathrm{Sx},x,q\right]\right):$
 > $P≔{\mathrm{Sx}}^{2}-x$
 ${P}{≔}{{\mathrm{Sx}}}^{{2}}{-}{x}$ (9)
 > $Q≔x\mathrm{Sx}$
 ${Q}{≔}{x}{}{\mathrm{Sx}}$ (10)
 > $\mathrm{skew_pdiv}\left(P,Q,\mathrm{Sx},A\right)$
 $\left[{q}{}{x}{,}{\mathrm{Sx}}{,}{-}{q}{}{{x}}^{{2}}\right]$ (11)
 > $\mathrm{skew_prem}\left(P,Q,\mathrm{Sx},A\right)$
 ${-}{q}{}{{x}}^{{2}}$ (12)

skew_elim (or skew_gcdex) may help to find factorization.

 > $P≔\left({q}^{2}x-1\right){\mathrm{Sx}}^{2}+\left({q}^{3}{x}^{2}+1+q-{q}^{2}x\right)\mathrm{Sx}-q$
 ${P}{≔}\left({{q}}^{{2}}{}{x}{-}{1}\right){}{{\mathrm{Sx}}}^{{2}}{+}\left({{q}}^{{3}}{}{{x}}^{{2}}{-}{{q}}^{{2}}{}{x}{+}{q}{+}{1}\right){}{\mathrm{Sx}}{-}{q}$ (13)
 > $Q≔\left({q}^{5}x+1\right)\left({q}^{5}x-1\right)\left({q}^{9}{x}^{2}-1\right){\mathrm{Sx}}^{5}+\left(-q+{q}^{12}{x}^{2}+{x}^{5}{q}^{23}-{q}^{4}-{q}^{2}+{x}^{2}{q}^{10}+{x}^{2}{q}^{11}-{q}^{3}+{q}^{22}{x}^{5}-1\right){\mathrm{Sx}}^{4}+q\left({x}^{2}{q}^{10}+{q}^{9}{x}^{2}+q+1+{q}^{6}+{q}^{16}{x}^{4}+2{q}^{4}+2{q}^{3}+{q}^{17}{x}^{4}+{q}^{5}+{q}^{24}{x}^{6}+{q}^{8}{x}^{2}+2{q}^{2}\right){\mathrm{Sx}}^{3}-{q}^{3}\left(1+{q}^{5}+{x}^{2}{q}^{10}+{x}^{2}{q}^{11}+{q}^{14}{x}^{4}+q+{q}^{7}{x}^{2}+{q}^{6}+{q}^{16}{x}^{4}+2{q}^{4}+2{q}^{3}+2{q}^{2}+{q}^{15}{x}^{4}+2{q}^{8}{x}^{2}+2{q}^{9}{x}^{2}\right){\mathrm{Sx}}^{2}+{q}^{6}\left({q}^{7}{x}^{2}+q+{q}^{3}+{q}^{6}{x}^{2}+1+{q}^{4}+{q}^{2}+{q}^{8}{x}^{2}\right)\mathrm{Sx}-{q}^{10}$
 ${Q}{≔}\left({{q}}^{{5}}{}{x}{+}{1}\right){}\left({{q}}^{{5}}{}{x}{-}{1}\right){}\left({{q}}^{{9}}{}{{x}}^{{2}}{-}{1}\right){}{{\mathrm{Sx}}}^{{5}}{+}\left({{x}}^{{5}}{}{{q}}^{{23}}{+}{{q}}^{{22}}{}{{x}}^{{5}}{+}{{q}}^{{12}}{}{{x}}^{{2}}{+}{{x}}^{{2}}{}{{q}}^{{11}}{+}{{x}}^{{2}}{}{{q}}^{{10}}{-}{{q}}^{{4}}{-}{{q}}^{{3}}{-}{{q}}^{{2}}{-}{q}{-}{1}\right){}{{\mathrm{Sx}}}^{{4}}{+}{q}{}\left({{q}}^{{24}}{}{{x}}^{{6}}{+}{{q}}^{{17}}{}{{x}}^{{4}}{+}{{q}}^{{16}}{}{{x}}^{{4}}{+}{{x}}^{{2}}{}{{q}}^{{10}}{+}{{q}}^{{9}}{}{{x}}^{{2}}{+}{{q}}^{{8}}{}{{x}}^{{2}}{+}{{q}}^{{6}}{+}{{q}}^{{5}}{+}{2}{}{{q}}^{{4}}{+}{2}{}{{q}}^{{3}}{+}{2}{}{{q}}^{{2}}{+}{q}{+}{1}\right){}{{\mathrm{Sx}}}^{{3}}{-}{{q}}^{{3}}{}\left({{q}}^{{16}}{}{{x}}^{{4}}{+}{{q}}^{{15}}{}{{x}}^{{4}}{+}{{q}}^{{14}}{}{{x}}^{{4}}{+}{{x}}^{{2}}{}{{q}}^{{11}}{+}{{x}}^{{2}}{}{{q}}^{{10}}{+}{2}{}{{q}}^{{9}}{}{{x}}^{{2}}{+}{2}{}{{q}}^{{8}}{}{{x}}^{{2}}{+}{{q}}^{{7}}{}{{x}}^{{2}}{+}{{q}}^{{6}}{+}{{q}}^{{5}}{+}{2}{}{{q}}^{{4}}{+}{2}{}{{q}}^{{3}}{+}{2}{}{{q}}^{{2}}{+}{q}{+}{1}\right){}{{\mathrm{Sx}}}^{{2}}{+}{{q}}^{{6}}{}\left({{q}}^{{8}}{}{{x}}^{{2}}{+}{{q}}^{{7}}{}{{x}}^{{2}}{+}{{q}}^{{6}}{}{{x}}^{{2}}{+}{{q}}^{{4}}{+}{{q}}^{{3}}{+}{{q}}^{{2}}{+}{q}{+}{1}\right){}{\mathrm{Sx}}{-}{{q}}^{{10}}$ (14)
 > $\mathrm{skew_elim}\left(P,Q,\mathrm{Sx},A\right)$
 ${\mathrm{Sx}}{}{{q}}^{{3}}{}{{x}}^{{2}}{+}{{\mathrm{Sx}}}^{{2}}{}{{q}}^{{2}}{}{x}{-}{\mathrm{Sx}}{}{{q}}^{{2}}{}{x}{-}{{\mathrm{Sx}}}^{{2}}{+}{q}{}{\mathrm{Sx}}{+}{\mathrm{Sx}}{-}{q}$ (15)

This is P. P therefore divides Q in A. Left gcds:

 > $A≔\mathrm{diff_algebra}\left(\left[\mathrm{Dx},x\right]\right):$
 > $L≔{\mathrm{Dx}}^{2}+1:$
 > $\mathrm{R1}≔x\mathrm{Dx}+1:$
 > $\mathrm{R2}≔{\mathrm{Dx}}^{2}+1:$
 > $\mathrm{P1}≔\mathrm{skew_product}\left(L,\mathrm{R1},A\right)$
 ${\mathrm{P1}}{≔}{{\mathrm{Dx}}}^{{3}}{}{x}{+}{3}{}{{\mathrm{Dx}}}^{{2}}{+}{\mathrm{Dx}}{}{x}{+}{1}$ (16)
 > $\mathrm{P2}≔\mathrm{skew_product}\left(L,\mathrm{R2},A\right)$
 ${\mathrm{P2}}{≔}{{\mathrm{Dx}}}^{{4}}{+}{2}{}{{\mathrm{Dx}}}^{{2}}{+}{1}$ (17)
 > $\mathrm{skew_gcdex}\left(\mathrm{P1},\mathrm{P2},\mathrm{Dx},A,"left"\right)$
 $\left[{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{2}}{+}{4}{}{\mathrm{Dx}}{}{x}{+}{{x}}^{{2}}{+}{2}{,}{-}{\mathrm{Dx}}{}{x}{-}{2}{,}{{x}}^{{2}}{,}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{2}}{+}{6}{}{\mathrm{Dx}}{}{x}{+}{{x}}^{{2}}{+}{6}{,}{-}{\mathrm{Dx}}{}{{x}}^{{3}}{-}{3}{}{{x}}^{{2}}\right]$ (18)
 > $\left[1\right]$
 ${{\mathrm{Dx}}}^{{2}}{}{{x}}^{{2}}{+}{4}{}{\mathrm{Dx}}{}{x}{+}{{x}}^{{2}}{+}{2}$ (19)
 > $\mathrm{skew_gcdex}\left(\mathrm{P1},\mathrm{P2},\mathrm{Dx},A,"left_monic"\right)$
 $\left[{{\mathrm{Dx}}}^{{2}}{+}{1}{,}{-}\frac{{\mathrm{Dx}}}{{x}}{,}{1}{,}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{2}}{+}{6}{}{\mathrm{Dx}}{}{x}{+}{{x}}^{{2}}{+}{6}{,}{-}{\mathrm{Dx}}{}{{x}}^{{3}}{-}{3}{}{{x}}^{{2}}\right]$ (20)
 > $\left[1\right]$
 ${{\mathrm{Dx}}}^{{2}}{+}{1}$ (21)