W-Gröbner basis and monomial ideals under polynomial composition

Abstract

The notion of weakly relatively prime and W-Grobner basis in K[x1, x2, …, xn] are given. The following results are obtained: for polynomials f1, f2, …, fm, \(\{ f_1^{\lambda _1 } ,f_2^{\lambda _2 } ,...,f_m^{\lambda _m } \} \) is a Grobner basis if and only if f1, f2, …, fm are pairwise weakly relatively prime with λ1, λ2, …, λm arbitrary non-negative integers; polynomial composition by Θ = (θ1, θ2, …, θn) commutes with monomial-Grobner bases computation if and only if θ1, θ2, …, θm are pairwise weakly relatively prime.