The dual_algebra(A, x_set)  function returns an Ore algebra $A^{\mathrm{`*`}}$ that is isomorphic to the opposite algebra $A^{\mathrm{op}}$ of A, that is, where the product $pq$ is defined as the value of the product $pq$ in A.

The dual_polynomial(p, A, x_set) function maps the polynomial p from A to a polynomial $p^{\mathrm{`*`}}$ in $A^{\mathrm{`*`}}$ so as to make the operator $\mathrm{`*`}$ an anti-isomorphism.  In other words, this operator follows the rule ${\left(pq\right)}^{\mathrm{`*`}}=q^{\mathrm{`*`}}{p}^{\mathrm{`*`}}$.

Both commands are useful to compute left gcds and to perform other calculations based on left skew Euclidean division (see examples below and skew_gcdex).

Skew polynomials of an Ore algebra A in the indeterminates $x_1,`...`,x_r,d_1,`...`,d_r$ (see skew_algebra) are represented under the normal form where all the x[i]s stand on the left of the monomials and all the d[i]s on the right.

The reverse_polynomial(p, A, x_set) function changes the representation of a skew polynomial p in A by moving all the d[i]s in x_set to the left of monomials, and the corresponding x[i]s to the right.

Correspondingly, the reverse_algebra(A, x_set) function returns an Ore algebra in which calculations with the new normal forms (returned by reverse_polynomial take place.

These functions are part of the Ore_algebra package, and so can be used in the form dual_algebra(..), dual_polynomial(..), reverse_algebra(..) or reverse_polynomial(..) 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[dual_algebra](..), Ore_algebra[dual_polynomial](..), Ore_algebra[reverse_algebra](..) and Ore_algebra[reverse_polynomial](..).

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

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

$\mathrm{dual\_polynomial}\left(x\&comma;A\&comma;\left\{\mathrm{Dx}\right\}\right)=\mathrm{reverse\_polynomial}\left(x\&comma;A\&comma;\left\{\mathrm{Dx}\right\}\right)$

$x=x$

$\mathrm{dual\_polynomial}\left(\mathrm{Dx}\&comma;A\&comma;\left\{\mathrm{Dx}\right\}\right)=\mathrm{reverse\_polynomial}\left(\mathrm{Dx}\&comma;A\&comma;\left\{\mathrm{Dx}\right\}\right)$

$\mathrm{Dx}=\mathrm{Dx}$

$p≔\mathrm{rand\_skew\_poly}\left(\left[x\&comma;\mathrm{Dx}\right]\&comma;A\right)$

$p≔-10{\mathrm{Dx}}^4+\left(-83x^2-73\right){\mathrm{Dx}}^2-4x^3\mathrm{Dx}+97x^2-62x$

$\mathrm{dual\_polynomial}\left(p\&comma;A\&comma;\left\{\mathrm{Dx}\right\}\right)$

$109x^2-62x-166-10{\mathrm{Dx}}^4+\left(-83x^2-73\right){\mathrm{Dx}}^2+\left(-4x^3+332x\right)\mathrm{Dx}$

$\mathrm{reverse\_polynomial}\left(p\&comma;A\&comma;\left\{\mathrm{Dx}\right\}\right)$

$-10{\mathrm{Dx}}^4-73{\mathrm{Dx}}^2-166-4x^3\mathrm{Dx}+\left(-83{\mathrm{Dx}}^2+109\right)x^2+\left(332\mathrm{Dx}-62\right)x$

$A≔\mathrm{skew\_algebra}\left(\mathrm{shift}=\left[\mathrm{Sn}\&comma;n\right]\right)\&colon;$

$\mathrm{dual\_polynomial}\left(n\&comma;A\&comma;\left\{\mathrm{Sn}\right\}\right)=\mathrm{reverse\_polynomial}\left(n\&comma;A\&comma;\left\{\mathrm{Sn}\right\}\right)$

$n=n$

$\mathrm{dual\_polynomial}\left(\mathrm{Sn}\&comma;A\&comma;\left\{\mathrm{Sn}\right\}\right)=\mathrm{reverse\_polynomial}\left(\mathrm{Sn}\&comma;A\&comma;\left\{\mathrm{Sn}\right\}\right)$

$\mathrm{Sn}=\mathrm{Sn}$

$p≔\mathrm{rand\_skew\_poly}\left(\left[n\&comma;\mathrm{Sn}\right]\&comma;A\right)$

$p≔74n{\mathrm{Sn}}^4+\left(6n^2+75n\right){\mathrm{Sn}}^3-92n^3{\mathrm{Sn}}^2+23n^4-50n$

$\mathrm{dual\_polynomial}\left(p\&comma;A\&comma;\left\{\mathrm{Sn}\right\}\right)$

$23n^4-50n+\left(74n-296\right){\mathrm{Sn}}^4+\left(6n^2+39n-171\right){\mathrm{Sn}}^3+\left(-92n^3+552n^2-1104n+736\right){\mathrm{Sn}}^2$

$\mathrm{reverse\_polynomial}\left(p\&comma;A\&comma;\left\{\mathrm{Sn}\right\}\right)$

$-296{\mathrm{Sn}}^4-171{\mathrm{Sn}}^3+736{\mathrm{Sn}}^2+23n^4-92n^3{\mathrm{Sn}}^2+\left(6{\mathrm{Sn}}^3+552{\mathrm{Sn}}^2\right)n^2+\left(74{\mathrm{Sn}}^4+39{\mathrm{Sn}}^3-1104{\mathrm{Sn}}^2-50\right)n$

$A≔\mathrm{skew\_algebra}\left(\mathrm{euler}=\left[\mathrm{Tx}\&comma;x\right]\right)\&colon;$

$\mathrm{dual\_polynomial}\left(x\&comma;A\&comma;\left\{\mathrm{Tx}\right\}\right)=\mathrm{reverse\_polynomial}\left(x\&comma;A\&comma;\left\{\mathrm{Tx}\right\}\right)$

$x=x$

$\mathrm{dual\_polynomial}\left(\mathrm{Tx}\&comma;A\&comma;\left\{\mathrm{Tx}\right\}\right)=\mathrm{reverse\_polynomial}\left(\mathrm{Tx}\&comma;A\&comma;\left\{\mathrm{Tx}\right\}\right)$

$\mathrm{Tx}=\mathrm{Tx}$

$p≔\mathrm{rand\_skew\_poly}\left(\left[x\&comma;\mathrm{Tx}\right]\&comma;A\right)$

$p≔\left(-29x^2-61x+10\right){\mathrm{Tx}}^2-8x^3\mathrm{Tx}+95x^5-23$

$\mathrm{dual\_polynomial}\left(p\&comma;A\&comma;\left\{\mathrm{Tx}\right\}\right)$

$95x^5+24x^3-116x^2-61x-23+\left(-29x^2-61x+10\right){\mathrm{Tx}}^2+\left(-8x^3+116x^2+122x\right)\mathrm{Tx}$

$\mathrm{reverse\_polynomial}\left(p\&comma;A\&comma;\left\{\mathrm{Tx}\right\}\right)$

$95x^5+\left(-8\mathrm{Tx}+24\right)x^3+\left(-29{\mathrm{Tx}}^2+116\mathrm{Tx}-116\right)x^2+\left(-61{\mathrm{Tx}}^2+122\mathrm{Tx}-61\right)x+10{\mathrm{Tx}}^2-23$

$A≔\mathrm{skew\_algebra}\left(\mathrm{qshift}=\left[\mathrm{Sn}\&comma;q^n\right]\right)\&colon;$

$\mathrm{dual\_polynomial}\left(q^n\&comma;A\&comma;\left\{\mathrm{Sn}\right\}\right)$

$q^n$

$\mathrm{reverse\_polynomial}\left(q^n\&comma;A\&comma;\left\{\mathrm{Sn}\right\}\right)$

Error, (in `index/Ore_algebra/should_not_be_used`) reverse not available for q-calculus algebras

$\mathrm{dual\_polynomial}\left(\mathrm{Sn}\&comma;A\&comma;\left\{\mathrm{Sn}\right\}\right)$

$\mathrm{Sn}$

$\mathrm{reverse\_polynomial}\left(\mathrm{Sn}\&comma;A\&comma;\left\{\mathrm{Sn}\right\}\right)$

Error, (in `index/Ore_algebra/should_not_be_used`) reverse not available for q-calculus algebras

$p≔\mathrm{rand\_skew\_poly}\left(\left[q^n\&comma;\mathrm{Sn}\right]\&comma;A\right)$

$p≔{\mathrm{Sn}}^5+77{\mathrm{Sn}}^4+q^n\left(95q^n-51\right){\mathrm{Sn}}^3+{\left(q^n\right)}^2\left(31q^n-10\right)\mathrm{Sn}$

$\mathrm{dual\_polynomial}\left(p\&comma;A\&comma;\left\{\mathrm{Sn}\right\}\right)$

${\mathrm{Sn}}^5+77{\mathrm{Sn}}^4+\frac{q^n\left(-51q^3+95q^n\right){\mathrm{Sn}}^3}{q^6}+\frac{{\left(q^n\right)}^2\left(31q^n-10q\right)\mathrm{Sn}}{q^3}$

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

$p≔\mathrm{rand\_skew\_poly}\left(\left[x\&comma;\mathrm{Dx}\right]\&comma;\mathrm{degree}=2\&comma;A\right)$

$p≔-27{\mathrm{Dx}}^2+\left(30x-28\right)\mathrm{Dx}+16x^2+55x+1$

$q≔\mathrm{rand\_skew\_poly}\left(\left[x\&comma;\mathrm{Dx}\right]\&comma;\mathrm{degree}=2\&comma;A\right)$

$q≔47{\mathrm{Dx}}^2+\left(-87x-96\right)\mathrm{Dx}+72x^2-59x-15$

$r≔\mathrm{rand\_skew\_poly}\left(\left[x\&comma;\mathrm{Dx}\right]\&comma;\mathrm{degree}=2\&comma;A\right)$

$r≔-48{\mathrm{Dx}}^2+\left(-88x+92\right)\mathrm{Dx}-91x^2+43x-90$

$P≔\mathrm{skew\_product}\left(r\&comma;p\&comma;A\right)$

$P≔1296{\mathrm{Dx}}^4+\left(936x-1140\right){\mathrm{Dx}}^3+\left(-951x^2+1423x-3074\right){\mathrm{Dx}}^2+\left(-4138x^3+470x^2-4644x+92\right)\mathrm{Dx}-1456x^4-4317x^3-1982x^2-6803x+3434$

$Q≔\mathrm{skew\_product}\left(r\&comma;q\&comma;A\right)$

$Q≔-2256{\mathrm{Dx}}^4+\left(40x+8932\right){\mathrm{Dx}}^3+\left(-77x^2+5297x-3990\right){\mathrm{Dx}}^2+\left(1581x^3+16811x^2-6574x+4920\right)\mathrm{Dx}-6552x^4+8465x^3-20324x^2+23105x-10990$

$\mathrm{dP}≔\mathrm{dual\_polynomial}\left(P\&comma;A\&comma;''fully''\right)$

$\mathrm{dP}≔-1456x^4-4317x^3+10432x^2-7743x+6176+\left(936x-1140\right){\mathrm{Dx}}^3+1296{\mathrm{Dx}}^4+\left(-951x^2+1423x-5882\right){\mathrm{Dx}}^2+\left(-4138x^3+470x^2-840x-2754\right)\mathrm{Dx}$

$\mathrm{dQ}≔\mathrm{dual\_polynomial}\left(Q\&comma;A\&comma;''fully''\right)$

$\mathrm{dQ}≔-6552x^4+8465x^3-25067x^2-10517x-4570+\left(40x+8932\right){\mathrm{Dx}}^3-2256{\mathrm{Dx}}^4+\left(-77x^2+5297x-4110\right){\mathrm{Dx}}^2+\left(1581x^3+16811x^2-6266x-5674\right)\mathrm{Dx}$

$\mathrm{dA}≔\mathrm{dual\_algebra}\left(A\&comma;''fully''\right)\&colon;$

$\mathrm{dGCD}≔\mathrm{skew\_gcdex}\left(\mathrm{dP}\&comma;\mathrm{dQ}\&comma;\mathrm{Dx}\&comma;\mathrm{dA}\right)$

$\mathrm{dGCD}≔\left[-8873046528{\mathrm{Dx}}^2{x}^4-16267251968\mathrm{Dx}{x}^5-16821817376x^6+24619089552{\mathrm{Dx}}^2{x}^3+62141670024\mathrm{Dx}{x}^4+54622461457x^5+32688986544{\mathrm{Dx}}^2{x}^2+12743220356x^3\mathrm{Dx}+39548558994x^4+92684791104{\mathrm{Dx}}^{2}x+107268226148\mathrm{Dx}{x}^2+147456828087x^3-35565352704{\mathrm{Dx}}^2-242848996240\mathrm{Dx}x-149094065426x^2+68166926016\mathrm{Dx}+35722494760x-1481889696\&comma;-14275216-\left(-2074251x-8632772\right)\mathrm{Dx}+2115893x^2-18025252x\&comma;2097234x^2-3015708x+7071808-\left(-1191591x-4959252\right)\mathrm{Dx}\&comma;-184855136{\mathrm{Dx}}^2{x}^4+342178656\mathrm{Dx}{x}^5-283182336x^6+512897699{\mathrm{Dx}}^2{x}^3-571829931\mathrm{Dx}{x}^4+1017767816x^5+681020553{\mathrm{Dx}}^2{x}^2-3047653889x^3\mathrm{Dx}+800590201x^4+1930933148{\mathrm{Dx}}^{2}x-3426608115\mathrm{Dx}{x}^2+2633091576x^3-740944848{\mathrm{Dx}}^2-1210456350\mathrm{Dx}x-9729185864x^2+3444352412\mathrm{Dx}+3624982240x-5687084608\&comma;-106193376{\mathrm{Dx}}^2{x}^4+117992640\mathrm{Dx}{x}^5+62929408x^6+294643359{\mathrm{Dx}}^2{x}^3-437507974\mathrm{Dx}{x}^4+41716368x^5+391224573{\mathrm{Dx}}^2{x}^2-553911398x^3\mathrm{Dx}-710110491x^4+1109259468{\mathrm{Dx}}^{2}x+57133929\mathrm{Dx}{x}^2-2374836150x^3-425649168{\mathrm{Dx}}^2+2405735818\mathrm{Dx}x-3008916049x^2+667845516\mathrm{Dx}+3496133096x+973816096\right]$

$\mathrm{dual\_polynomial}\left(\mathrm{dGCD}\left[1\right]\&comma;\mathrm{dA}\&comma;''fully''\right)$

$-16821817376x^6+54622461457x^5-41787700846x^4+396023508183x^3-217340962694x^2+397973484368x-178952912848+\left(-8873046528x^4+24619089552x^3+32688986544x^2+92684791104x-35565352704\right){\mathrm{Dx}}^2+\left(-16267251968x^5+62141670024x^4-58241151868x^3+254982763460x^2-112093050064x+253536508224\right)\mathrm{Dx}$

$\mathrm{lgcd}≔\mathrm{skew\_product}\left(\&comma;\frac{1}{\mathrm{lcoeff}\left(\&comma;\mathrm{Dx}\right)}\&comma;A\right)$

$\mathrm{lgcd}≔\frac{91x^2}{48}-\frac{43x}{48}+\frac{15}{8}+{\mathrm{Dx}}^2+\left(\frac{11x}{6}-\frac{23}{12}\right)\mathrm{Dx}$

$A\left[''normalizer''\right]\left(\frac{r}{\mathrm{lcoeff}\left(r\&comma;\mathrm{Dx}\right)}\right)$

$\frac{91x^2}{48}-\frac{43x}{48}+\frac{15}{8}+{\mathrm{Dx}}^2+\left(\frac{11x}{6}-\frac{23}{12}\right)\mathrm{Dx}$

$\mathrm{skew\_gcdex}\left(P\&comma;Q\&comma;\mathrm{Dx}\&comma;A\&comma;''left\_monic''\right)\left[1\right]$

$\frac{91x^2}{48}-\frac{43x}{48}+\frac{15}{8}+{\mathrm{Dx}}^2+\left(\frac{11x}{6}-\frac{23}{12}\right)\mathrm{Dx}$