Reduced Gröbner Bases Under Composition

In this paper we contribute with one main result to the interesting problem initiated by Hong (1998, J. Symb. Comput.25, 643–663) on the behaviour of Gröbner bases under composition of polynomials. Polynomial composition is the operation of replacing the variables of a polynomial with other polynomials. The main question of this paper is:When does composition commute with reduced Gröbner bases computation under the same term ordering?We give a complete answer for this question: let Θ be a polynomial map, then for every reduced GröbnerbasisG,G ○ Θ is a reduced Gröbner basis if and only if the composition by Θ is compatible with the term ordering and Θ is a list of permuted univariate and monic polynomials. Besides, we also include other minor results concerned with this problem; in particular, we provide a sufficient condition to determine when composition commutes with reduced Gröbner bases computation (possibly) under different term ordering.