Groebner Basis Under Composition I

Composition is the operation of replacing variables in a polynomial with other polynomials. The main question of this paper is:When does composition commute with Groebner basis computation?We prove that this happens iff the composition is ‘compatible’ with the term ordering and the nondivisibility. This has a natural application in the computation of Groebner bases of composed polynomials which often arises in real-life problems.