dual_algebra(A, x_set)

dual_polynomial(p, A, x_set)

reverse_algebra(A, x_set)

reverse_polynomial(p, A, x_set)

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.

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.

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.

`q`-Shift operators

Only dual polynomials are available.

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.

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.

The dual of a dual polynomial is the polynomial.

This is the left gcd, up to renormalization (by multiplication by a rational function on the right).

This is also the built-in left factor r, up to renormalization (by multiplication by a rational function on the right).

This calculation is that performed by Ore_algebra[skew_gcdex] with the options  and .

Ore_algebra, Ore_algebra/skew_algebra, Ore_algebra/skew_gcdex