Computing Elements of Certain Form in Ideals to Prove Properties of Operators

Abstract Proving statements about linear operators expressed in terms of identities often leads to finding elements certain form noncommutative polynomial ideals. We illustrate this by examples coming from actual operator and discuss relevant algorithmic methods for such polynomials based on Gröbner bases. In particular, we present algorithms computing the intersection a two-sided ideal with one-sided as well homogeneous ideals monomials All presented work are implemented Mathematica package .