Univariate Ore Polynomial Rings in Computer Algebra

Radicals of ore polynomials

“…For an Ore algebra K[y][X;σ,δ] with nontrivial σ and δ it is well known that there exists a computable isomorphism φ from K(y)[X;σ,δ] to the algebra K(y)[X;σ,0] with trivial pseudo-derivation [1]. Starting with P ∈ K[y][X;σ,δ], it might happen that φ(P ) has rational function coefficients.…”

Submersive rational difference systems and their accessibility

“…Thus,σ (r) belongs to τ(K). By Theorem II in [8, page 66], K is the inversive closure of τ(K), and so is that of K. 2 Assume now that J is a difference ideal associated with a submersive system Σ given by (1). By Propositions 3.4 and 4.2, the field K Σ associated with Σ has an inversive closure, which is called the inversive field associated with Σ, and is denoted by K Σ ,σ .…”

“…y [4] = y [0] y [1] u [2] + u [0] u [1] y [3] + u [0] y [2] u [3] ( [2] s − y [1] u [2] , Q = u [0] y [2] s 3 + y [0] y [1] s 2 + u [0] y [3] s + u [1] y [3] + y [2] u [3] . We cannot assert that D is zero when a homomorphic image of D vanishes.…”

“…Their applications, in general, require the help of computer algebra software. In Maple there exists a built-in package, OreTools [1], that addresses the computations with Ore polynomials. Unfortunately, this package, though being a good starting point for control applications, cannot handle the Ore polynomials associated to nonlinear control systems without further extension.…”

Practical polynomial formulas in MIMO nonlinear realization problem

“…. , m) T , and 1 The form (4) is an extension of the Guidorzi canonical form, introduced in [3] for linear systems and typically applied in system identification. The other forms, like Hermite or Popov form, may be also used.…”