Conditional GrÖbner Basis: GrÖbner Basis Detection with Parameters

For a given finite polynomial set, finding a monomial order such that the given set is already a GrÖbner basis for the ideal generated by the given set with respect to the found monomial order is called GrÖbner basis detection (GBD) problem and there is also its simpler version, called structural GrÖbner basis detection (SGBD) problem. In this short communication, we give algorithms to solve these problems for polynomials with parameters on their coefficients.