Abstract | Let K [ x 1 , … , x n ] be the polynomial ring over a field K in variables x 1 , … , x n . Let Θ = ( θ 1 , … , θ n ) be a list of n homogeneous polynomials of same degree in K [ x 1 , … , x n ] . Polynomial composition by Θ is the operation of replacing x i of a polynomial by θ i . The main question of this paper is: When does homogeneous polynomial composition commute with homogeneous Gröbner bases computation under the same term ordering? We give a complete answer: for every homogeneous Gröbner basis G, G ○ Θ is a homogeneous Gröbner basis under the same term ordering if and only if the composition by Θ is homogeneously compatible with the term ordering and Θ is a ‘permuted powering.’ |