Polynomial Interpolation Problem for Skew Polynomials

Let R = K[x;σ] be a skew polynomial ring over a division ring K. We introduce the notion of derivatives of skew polynomial at scalars. An analogous definition of derivatives of commutative polynomials from K[x] as a function of K[x] → K[x] is not possible in a non-commutative case. This is the reason why we have to define the derivative of a skew polynomial at a scalar. Our definition is based on properties of skew polynomial rings, and it makes possible some useful theorems about them. The main result of this paper is a generalization of polynomial interpolation problem for skew polynomials. We present conditions under which there exists a unique polynomial of a degree less then n which takes prescribed values at given points xi ∈ K (1 ≤ n). We also discuss some kind of Silvester-Lagrange skew polynomial.