return the dual algebra of an Ore algebra, that is, its opposite ring

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Ore_algebra[dual_algebra] - return the dual algebra of an Ore algebra, that is, its opposite ring

Ore_algebra[dual_polynomial] - map a skew polynomial of an Ore algebra to the dual algebra

Ore_algebra[reverse_algebra] - return an Ore algebra with opposite normal forms

Ore_algebra[reverse_polynomial] - change normal form of a skew polynomial in an Ore algebra

	Calling Sequence
	dual_algebra(A, x_set) dual_polynomial(p, A, x_set) reverse_algebra(A, x_set) reverse_polynomial(p, A, x_set)

Parameters

A	-	Ore algebra
x_set	-	subset of the (polynomial) indeterminates of the algebra, or the string to denote all indeterminates
p	-	skew polynomial

Description

•	The dual_algebra(A, x_set) function returns an Ore algebra that is isomorphic to the opposite algebra of A, that is, where the product is defined as the value of the product in A.

•	The dual_polynomial(p, A, x_set) function maps the polynomial p from A to a polynomial in so as to make the operator an anti-isomorphism. In other words, this operator follows the rule .

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 (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](..).

Examples

Differential operators

Dual and reverse polynomials look similar, but the dual polynomial is a polynomial in Dx with coefficients in x while the reverse polynomial is a polynomial in x with coefficients in Dx.

>	$dual_polynomial(x, A, {Dx}) = reverse_polynomial(x, A, {Dx})$

(1)

>	$dual_polynomial(Dx, A, {Dx}) = reverse_polynomial(Dx, A, {Dx})$

(2)

(3)

>	$dual_polynomial(p, A, {Dx})$

(4)

>	$reverse_polynomial(p, A, {Dx})$

(5)

Shift operators

Dual and reverse polynomials look similar, but the dual polynomial is a polynomial in Sn with coefficients in n while the reverse polynomial is a polynomial in n with coefficients in Sn.

>	$dual_polynomial(n, A, {Sn}) = reverse_polynomial(n, A, {Sn})$

(6)

>	$dual_polynomial(Sn, A, {Sn}) = reverse_polynomial(Sn, A, {Sn})$

(7)

(8)

>	$dual_polynomial(p, A, {Sn})$

(9)

>	$reverse_polynomial(p, A, {Sn})$

(10)

Eulerian operators

Dual and reverse polynomials look similar, but the dual polynomial is a polynomial in Tx with coefficients in x while the reverse polynomial is a polynomial in x with coefficients in Tx.

>	$dual_polynomial(x, A, {Tx}) = reverse_polynomial(x, A, {Tx})$

(11)

>	$dual_polynomial(Tx, A, {Tx}) = reverse_polynomial(Tx, A, {Tx})$

(12)

(13)

>	$dual_polynomial(p, A, {Tx})$

(14)

>	$reverse_polynomial(p, A, {Tx})$

(15)

`q`-Shift operators

Only dual polynomials are available.

>	$dual_polynomial(q^n, A, {Sn})$

(16)

>	$reverse_polynomial(q^n, A, {Sn})$

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

>	$dual_polynomial(Sn, A, {Sn})$

(17)

>	$reverse_polynomial(Sn, A, {Sn})$

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

(18)

>	$dual_polynomial(p, A, {Sn})$

77*Sn^4+q^n*(95*q^n-51*q^3)*Sn^3/q^6+(q^n)^2*(31*q^n-10*q)*Sn/q^3+Sn^5

(19)

Computation of left gcds and left lcms

The function Ore_algebra[skew_gcdex] inputs two polynomials p and q and computes a list such that and . The polynomial g is a right gcd of p and q. Applying the dualization operator yields a list such that and , where is a left gcd of and . The following method to compute left gcds is based on this idea.

Define two polynomials P and Q that share a left common divisor.

(20)

(21)

(22)

(23)

(24)

Introduce their dual polynomials and compute their right gcd in the dual algebra, corresponding the to left gcd of the original polynomials in the original algebra.

(25)

(26)

(27)

The dual of a dual polynomial is the polynomial.

-178952912848+397973484368*x+54622461457*x^5-217340962694*x^2-41787700846*x^4+396023508183*x^3-16821817376*x^6+(-35565352704+32688986544*x^2+92684791104*x-8873046528*x^4+24619089552*x^3)*Dx^2+(254982763460*x^2-112093050064*x+253536508224-58241151868*x^3+62141670024*x^4-16267251968*x^5)*Dx