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  such that  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 , it returns a list  such that  and .  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  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  such that  and .  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  of skew polynomials of the algebra A such that .

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  free from x, is such a polynomial exists.  Otherwise, a nonzero polynomial  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](..).