Homogeneous Gröbner bases under composition

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.'

Further results on homogeneous Gröbner bases under composition

“…A complete characterization is given for homogeneous Gröbner bases under an arbitrary grading. This unifies the results of Hong (1998) and Liu and Wang (2006). …”

“…Notice that in Definition 2.5, we require that G • Θ is only a Gröbner basis. Thus we do not need to set a restriction on Θ as in [15]. Definition 2.6.…”

“…If Γ (x 1 ) = · · · = Γ (x n ) = 0, then Definition 2.6 is the same as Definition 2.3, and if Γ (x 1 ) = · · · = Γ (x n ) = 1, then it is just Definition 2.6 in [15].…”

“…In [15], the authors studied the problem of the behavior of homogeneous Gröbner bases in the usual sense under composition of polynomials. It is rather natural to ask if the theorem of Hong [8] and that of [15] can be unified under a more general framework.…”

“…The aim of the paper is to give a necessary and sufficient condition to the above problem, which may be regarded as a common generalization of Hong's theorem [8] and the result in [15].…”