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,<function>).  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](..).

$\mathrm{with}\left(\mathrm{Ore\_algebra}\right)\&colon;$

$A≔\mathrm{diff\_algebra}\left(\left[\mathrm{Dx}\&comma;x\right]\right)\&colon;$

$P≔\mathrm{skew\_product}\left(x\left({\mathrm{Dx}}^2\&plus;\mathrm{Dx}-1\right)\&comma;\left(xx-1\right)\left(\mathrm{Dx}\mathrm{Dx}\&plus;1\right)\&comma;A\right)\&colon;$

$Q≔\mathrm{skew\_product}\left(\left(x-1\right)\left({\mathrm{Dx}}^2-\mathrm{Dx}\&plus;1\right)\&comma;\left(xx-1\right)\left(\mathrm{Dx}\mathrm{Dx}\&plus;1\right)\&comma;A\right)\&colon;$

$\mathrm{skew\_pdiv}\left(P\&comma;Q\&comma;\mathrm{Dx}\&comma;A\right)$

$\left[x-1\&comma;x\&comma;2{\mathrm{Dx}}^3{x}^4-2{\mathrm{Dx}}^3{x}^3-2{\mathrm{Dx}}^2{x}^4-2{\mathrm{Dx}}^3{x}^2+6{\mathrm{Dx}}^2{x}^3+2\mathrm{Dx}{x}^4+2{\mathrm{Dx}}^{3}x-2{\mathrm{Dx}}^2{x}^2-2\mathrm{Dx}{x}^3-2x^4-2{\mathrm{Dx}}^{2}x-2\mathrm{Dx}{x}^2+6x^3+2\mathrm{Dx}x-2x^2-2x\right]$

$G≔\mathrm{skew\_gcdex}\left(P\&comma;Q\&comma;\mathrm{Dx}\&comma;A\right)$

$G≔\left[2{\mathrm{Dx}}^2{x}^6-4{\mathrm{Dx}}^2{x}^5+2x^6+4{\mathrm{Dx}}^2{x}^3-4x^5-2{\mathrm{Dx}}^2{x}^2+4x^3-2x^2\&comma;1-\left(x^3-2x^2+x\right)\mathrm{Dx}+x^2-2x\&comma;2x^3-3x^2-\left(-x^3+x^2\right)\mathrm{Dx}\&comma;{\mathrm{Dx}}^2{x}^5-3{\mathrm{Dx}}^2{x}^4-\mathrm{Dx}{x}^5+3{\mathrm{Dx}}^2{x}^3+\mathrm{Dx}{x}^4+x^5-{\mathrm{Dx}}^2{x}^2+3\mathrm{Dx}{x}^3-2x^4-5\mathrm{Dx}{x}^2+2x^3+2\mathrm{Dx}x-4x^2+5x-2\&comma;-{\mathrm{Dx}}^2{x}^5+2{\mathrm{Dx}}^2{x}^4-\mathrm{Dx}{x}^5-{\mathrm{Dx}}^2{x}^3+4\mathrm{Dx}{x}^4+x^5-3\mathrm{Dx}{x}^3-x^4-2x^3\right]$

$\mathrm{skew\_product}\left(G_2\&comma;P\&comma;A\right)\&plus;\mathrm{skew\_product}\left(G_3\&comma;Q\&comma;A\right)-G_1$

$-2{\mathrm{Dx}}^2{x}^6+4{\mathrm{Dx}}^2{x}^5+\left(-4x^5+2x^4+8x^3-6x^2\right){\mathrm{Dx}}^3+\left(3x^6-2x^5-6x^4+6x^3-x^2\right){\mathrm{Dx}}^2+\left(-x^6+6x^5-2x^4-10x^3+7x^2\right)\mathrm{Dx}+\left(x^6+4x^5-12x^4+8x^3-x^2\right){\mathrm{Dx}}^4+\left(x^6-2x^5+2x^3-x^2\right){\mathrm{Dx}}^5+\left(-x^6-2x^5+6x^4-2x^3-x^2\right){\mathrm{Dx}}^2+\left(-x^6-4x^5+12x^4-8x^3+x^2\right){\mathrm{Dx}}^4+\left(4x^5-2x^4-8x^3+6x^2\right){\mathrm{Dx}}^3+\left(x^6-6x^5+2x^4+10x^3-7x^2\right)\mathrm{Dx}+\left(-x^6+2x^5-2x^3+x^2\right){\mathrm{Dx}}^5-4{\mathrm{Dx}}^2{x}^3+2{\mathrm{Dx}}^2{x}^2$

$\mathrm{skew\_product}\left(G_4\&comma;P\&comma;A\right)\&plus;\mathrm{skew\_product}\left(G_5\&comma;Q\&comma;A\right)$

$\left(x^8-3x^7+2x^6+2x^5-3x^4+x^3\right){\mathrm{Dx}}^6+\left(8x^7-24x^6+24x^5-8x^4\right){\mathrm{Dx}}^5+\left(12x^6-36x^5+36x^4-12x^3\right){\mathrm{Dx}}^4+\left(2x^8-2x^7-8x^6+16x^5-10x^4+2x^3\right){\mathrm{Dx}}^3+\left(-2x^8+10x^7-6x^6-22x^5+32x^4-12x^3\right){\mathrm{Dx}}^2+\left(2x^8-10x^7+16x^6-8x^5-2x^4+2x^3\right)\mathrm{Dx}+\left(-x^8+3x^7-2x^6-2x^5+3x^4-x^3\right){\mathrm{Dx}}^6+\left(-8x^7+24x^6-24x^5+8x^4\right){\mathrm{Dx}}^5+\left(-12x^6+36x^5-36x^4+12x^3\right){\mathrm{Dx}}^4+\left(-2x^8+2x^7+8x^6-16x^5+10x^4-2x^3\right){\mathrm{Dx}}^3+\left(2x^8-10x^7+6x^6+22x^5-32x^4+12x^3\right){\mathrm{Dx}}^2+\left(-2x^8+10x^7-16x^6+8x^5+2x^4-2x^3\right)\mathrm{Dx}$

$\mathrm{annihilators}\left(P\&comma;Q\&comma;A\right)$

$\left[{\mathrm{Dx}}^2{x}^5-3{\mathrm{Dx}}^2{x}^4-\mathrm{Dx}{x}^5+3{\mathrm{Dx}}^2{x}^3+\mathrm{Dx}{x}^4+x^5-{\mathrm{Dx}}^2{x}^2+3\mathrm{Dx}{x}^3-2x^4-5\mathrm{Dx}{x}^2+2x^3+2\mathrm{Dx}x-4x^2+5x-2\&comma;-{\mathrm{Dx}}^2{x}^5+2{\mathrm{Dx}}^2{x}^4-\mathrm{Dx}{x}^5-{\mathrm{Dx}}^2{x}^3+4\mathrm{Dx}{x}^4+x^5-3\mathrm{Dx}{x}^3-x^4-2x^3\right]$

$A≔\mathrm{diff\_algebra}\left(\left[\mathrm{Dx}\&comma;x\right]\&comma;\mathrm{polynom}\&equals;x\right)\&colon;$

$G≔\mathrm{skew\_gcdex}\left(P\&comma;Q\&comma;x\&comma;A\right)$

$G≔\left[{\mathrm{Dx}}^6{x}^2-{\mathrm{Dx}}^6+8{\mathrm{Dx}}^{5}x+2{\mathrm{Dx}}^4{x}^2-2{\mathrm{Dx}}^3{x}^2+10{\mathrm{Dx}}^4+12{\mathrm{Dx}}^{3}x+2{\mathrm{Dx}}^3-4{\mathrm{Dx}}^{2}x-2\mathrm{Dx}{x}^2+14{\mathrm{Dx}}^2+4\mathrm{Dx}x-x^2+2\mathrm{Dx}-4x+3\&comma;{\mathrm{Dx}}^2-\mathrm{Dx}+1\&comma;-{\mathrm{Dx}}^2-\mathrm{Dx}+1\&comma;-{\mathrm{Dx}}^{8}x+{\mathrm{Dx}}^8+2{\mathrm{Dx}}^{7}x-10{\mathrm{Dx}}^7-4{\mathrm{Dx}}^{6}x+20{\mathrm{Dx}}^6+6{\mathrm{Dx}}^{5}x-34{\mathrm{Dx}}^5-7{\mathrm{Dx}}^{4}x+35{\mathrm{Dx}}^4+6{\mathrm{Dx}}^{3}x-34{\mathrm{Dx}}^3-2{\mathrm{Dx}}^{2}x+22{\mathrm{Dx}}^2-12\mathrm{Dx}+x+3\&comma;{\mathrm{Dx}}^{8}x+8{\mathrm{Dx}}^7+2{\mathrm{Dx}}^6-8{\mathrm{Dx}}^5-3{\mathrm{Dx}}^{4}x+14{\mathrm{Dx}}^4+4{\mathrm{Dx}}^{3}x-20{\mathrm{Dx}}^3-4{\mathrm{Dx}}^{2}x+18{\mathrm{Dx}}^2-8\mathrm{Dx}+x+4\right]$

$\mathrm{skew\_product}\left(G_2\&comma;P\&comma;A\right)\&plus;\mathrm{skew\_product}\left(G_3\&comma;Q\&comma;A\right)-G_1$

$0$

$\mathrm{skew\_product}\left(G_4\&comma;P\&comma;A\right)\&plus;\mathrm{skew\_product}\left(G_5\&comma;Q\&comma;A\right)$

$0$

$A≔\mathrm{skew\_algebra}\left(\mathrm{comm}\&equals;q\&comma;\mathrm{qdilat}\&equals;\left[\mathrm{Sx}\&comma;x\&comma;q\right]\right)\&colon;$

$P≔{\mathrm{Sx}}^2-x$

$P≔{\mathrm{Sx}}^2-x$

$Q≔x\mathrm{Sx}$

$Q≔x\mathrm{Sx}$

$\mathrm{skew\_pdiv}\left(P\&comma;Q\&comma;\mathrm{Sx}\&comma;A\right)$

$\left[qx\&comma;\mathrm{Sx}\&comma;-q{x}^2\right]$

$\mathrm{skew\_prem}\left(P\&comma;Q\&comma;\mathrm{Sx}\&comma;A\right)$

$-q{x}^2$

$P≔\left(q^{2}x-1\right){\mathrm{Sx}}^2\&plus;\left(q^3{x}^2\&plus;1\&plus;q-q^{2}x\right)\mathrm{Sx}-q$

$P≔\left(q^{2}x-1\right){\mathrm{Sx}}^2+\left(q^3{x}^2-q^{2}x+q+1\right)\mathrm{Sx}-q$

$Q≔\left(q^{5}x\&plus;1\right)\left(q^{5}x-1\right)\left(q^9{x}^2-1\right){\mathrm{Sx}}^5\&plus;\left(-q\&plus;q^{12}{x}^2\&plus;x^5{q}^{23}-q^4-q^2\&plus;x^2{q}^{10}\&plus;x^2{q}^{11}-q^3\&plus;q^{22}{x}^5-1\right){\mathrm{Sx}}^4\&plus;q\left(x^2{q}^{10}\&plus;q^9{x}^2\&plus;q\&plus;1\&plus;q^6\&plus;q^{16}{x}^4\&plus;2q^4\&plus;2q^3\&plus;q^{17}{x}^4\&plus;q^5\&plus;q^{24}{x}^6\&plus;q^8{x}^2\&plus;2q^2\right){\mathrm{Sx}}^3-q^3\left(1\&plus;q^5\&plus;x^2{q}^{10}\&plus;x^2{q}^{11}\&plus;q^{14}{x}^4\&plus;q\&plus;q^7{x}^2\&plus;q^6\&plus;q^{16}{x}^4\&plus;2q^4\&plus;2q^3\&plus;2q^2\&plus;q^{15}{x}^4\&plus;2q^8{x}^2\&plus;2q^9{x}^2\right){\mathrm{Sx}}^2\&plus;q^6\left(q^7{x}^2\&plus;q\&plus;q^3\&plus;q^6{x}^2\&plus;1\&plus;q^4\&plus;q^2\&plus;q^8{x}^2\right)\mathrm{Sx}-q^{10}$

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

$\mathrm{skew\_elim}\left(P\&comma;Q\&comma;\mathrm{Sx}\&comma;A\right)$

$\mathrm{Sx}{q}^3{x}^2+{\mathrm{Sx}}^2{q}^{2}x-\mathrm{Sx}{q}^{2}x-{\mathrm{Sx}}^2+q\mathrm{Sx}+\mathrm{Sx}-q$

$A≔\mathrm{diff\_algebra}\left(\left[\mathrm{Dx}\&comma;x\right]\right)\&colon;$

$L≔{\mathrm{Dx}}^2\&plus;1\&colon;$

$\mathrm{R1}≔x\mathrm{Dx}\&plus;1\&colon;$

$\mathrm{R2}≔{\mathrm{Dx}}^2\&plus;1\&colon;$

$\mathrm{P1}≔\mathrm{skew\_product}\left(L\&comma;\mathrm{R1}\&comma;A\right)$

$\mathrm{P1}≔{\mathrm{Dx}}^{3}x+3{\mathrm{Dx}}^2+\mathrm{Dx}x+1$

$\mathrm{P2}≔\mathrm{skew\_product}\left(L\&comma;\mathrm{R2}\&comma;A\right)$

$\mathrm{P2}≔{\mathrm{Dx}}^4+2{\mathrm{Dx}}^2+1$

$\mathrm{skew\_gcdex}\left(\mathrm{P1}\&comma;\mathrm{P2}\&comma;\mathrm{Dx}\&comma;A\&comma;''left''\right)$

$\left[{\mathrm{Dx}}^2{x}^2+4\mathrm{Dx}x+x^2+2\&comma;-\mathrm{Dx}x-2\&comma;x^2\&comma;{\mathrm{Dx}}^2{x}^2+6\mathrm{Dx}x+x^2+6\&comma;-\mathrm{Dx}{x}^3-3x^2\right]$

$1_{}$

${\mathrm{Dx}}^2{x}^2+4\mathrm{Dx}x+x^2+2$

$\mathrm{skew\_gcdex}\left(\mathrm{P1}\&comma;\mathrm{P2}\&comma;\mathrm{Dx}\&comma;A\&comma;''left\_monic''\right)$

$\left[{\mathrm{Dx}}^2+1\&comma;-\frac{\mathrm{Dx}}{x}\&comma;1\&comma;{\mathrm{Dx}}^2{x}^2+6\mathrm{Dx}x+x^2+6\&comma;-\mathrm{Dx}{x}^3-3x^2\right]$

$1_{}$

${\mathrm{Dx}}^2+1$