Groebner Basis Under Composition I

The main question of this paper is: When does Groebner basis computation (Buchberger, 1965, 1985) commute with composition? More precisely, let F be a finite set of polynomials in the variables x1, . . . , xn, and let G be a Groebner basis of the ideal generated by F under some term ordering. Let Θ = (θ1, . . . , θn) be a list of n polynomials in the variables x1, . . . , xn. Let F ∗ be the set obtained from F by replacing xi by θi and likewise let G∗ be the set obtained from G by replacing xi by θi. One ponders whether G∗ is also a Groebner basis of F ∗ (under the same term ordering). It is not. One can easily construct counterexamples (for instance, just permute the variables) but one can also find numerous positive examples. Thus, the following question naturally arises: When is G∗ a Groebner basis of F ∗? In other words, when does Groebner basis computation commute with composition? The main contribution of this paper is to show that Groebner basis computation commutes with composition iff the composition is ‘compatible’ with the term ordering and the nondivisibility. Apart from satisfying curiosity, the answer to such a question has a natural application in the computation of a Groebner basis of the ideal generated by composed polynomials. In order to compute a Groebner basis of F ∗, we first compute a Groebner basis G of F and carry out the composition on G, obtaining a Groebner basis of F ∗. This should be more efficient than computing a Groebner basis of F ∗ directly (ignoring the structural information). Composed objects (polynomials) often occur in real-life problem-solving because the