skew lcm of a pair of operators
extended skew gcd computation
skew elimination of an indeterminate
annihilators(p, q, A)
skew_gcdex(p, q, x, A, opt)
skew_pdiv(p, q, x, A)
skew_prem(p, q, x, A)
skew_elim(p, q, x, A)
Ore algebra table
indeterminate of the algebra
(optional) literal string; one of monic, left, and left_monic
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 u,v,r such that u⁢p−v⁢q=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 g,a,b,u,v such that up+vq=0 and ap+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 g,a,b,u,v such that pu+qv=0 and pa+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 u,v of skew polynomials of the algebra A such that up+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 ap+bq free from x, is such a polynomial exists. Otherwise, a nonzero polynomial ap+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](..).
The skew polynomials can be viewed as polynomials in Dx
or in x. In this case, the algebra must be redefined accordingly.
Case of 'q'-calculus:
skew_elim (or skew_gcdex) may help to find factorization.
This is P. P therefore divides Q in A. Left gcds:
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