Abstract
We consider the factorization problems of difference operators in $\mathbb{C}[x;\sigma]$ for an automorphism σ of finite order. We study the factorization of regular polynomials in $\mathbb{R}[x]$ in the ring of such difference operators and obtain an analogue of the fundamental theorem of algebra for skew polynomial ring $\mathsf{K}[x; \sigma]$ over field K.
Highlights
Let R be a ring, σ an endomorphism of R, and δ a left σ -derivation of R
The skew polynomial ring R[x; σ, δ] is the set of all polynomials a + a x + · · · + anxn, where a, . . . , an ∈ R, addition is defined as usual, and multiplication is defined by xa = σ (a)x + δ(a) for all a ∈ R
Linear differential operators (σ = idR, the identity map on R) and linear difference operators (δ =, the identically zero function) are special cases of above skew polynomials, which have been studied via algebraic methods since [ ]
Summary
Let R be a ring, σ an endomorphism of R, and δ a left σ -derivation of R. Factorization of skew polynomials is a recent active area in computer algebra (see, for example, [ – ]). Throughout this paper, we assume that R[x; σ ] is a skew polynomial ring with automorphism σ as defined above.
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