On the structure of semi-invariant polynomials in Ore extensions

Let D be an X-outer S-derivation of a prime ring R, where S is an automorphism of R. The following is proved among other things: The degree of the minimal semi-invariant polynomial of the Ore extension R[X;S,D] is ν if charR=0, and is pkν for some k⩾0 if charR=p⩾2, where ν is the least integer ν⩾1 such that SνDS−ν−D is X-inner. A similar result holds for cv-polynomials. These are done by introducing the new notion of k-basic polynomials for each integer k⩾0, which enable us to analyze semi-invariant polynomials inductively.