Parametric FGLM algorithm

In this paper, we consider the non-trivial problem of converting a zero-dimensional parametric Gröbner basis w.r.t. a given monomial ordering to a Gröbner basis w.r.t. any other monomial ordering. We present a new algorithm, so-called parametric FGLM algorithm, that takes as input a monomial ordering and a finite parametric set which is a Gröbner basis w.r.t. a given set of parametric constraints, and outputs a decomposition of the given space of parameters as a finite set of (parametric) cells and for each cell a finite set of parametric polynomials which is a Gröbner basis w.r.t. the target monomial ordering and the corresponding cell. For this purpose, we develop computationally efficient algorithms to deal with parametric linear systems that are applicable in computing comprehensive Gröbner systems of parametric linear ideals also in the theory of parametric linear algebra to compute Gaussian elimination and minimal polynomial of a parametric matrix. All proposed algorithms have been implemented in Maple and their efficiency is discussed on a diverse set of benchmark polynomials.