When does \(\langle T \rangle\) equal \(\text{sat}(T)\)?
Let \(S=\Bbbk[x_1,\dots,x_n]\) be a multivariate polynomial ring with ordered variables \(x_1\prec x_2 \prec \dots \prec x_n\). For any polynomial \(p\in S\) the largest variable dividing one of its terms is the \textit{main variable} \(\text{mvar}(p)\) of \(p\). Then \(p\) is regarded as a univariate polynomial in \(\text{mvar}(p)\) and its leading coefficient is denoted \(\text{init}(p)\). For any set \(T\subset S\) define \(\text{sat} (T) := \langle T\rangle : h^\infty\) where \(h\) is the product of all \(\text{init}(t)\), \(t\in T\). A \textit{regular chain} is defined inductively. The empty set is a regular chain. If \(T\) is a set of polynomials, then write \(T = \{p\} \cup T'\) where \(p\in T\) has the greatest main variable among the elements of \(T\). In this case \(T\) is a regular chain if \(T'\) is and \(\text{init}(p)\) is regular modulo \(\text{sat}(T')\). For example, every regular sequence is a regular chain, but not the other way around. The main use of regular chains is that the containments \(p \in \text{sat}(T)\) and \(p\in \sqrt{\text{sat}(T)}\) can be tested efficiently, that is without computing a Gröbner basis of \(\langle T \rangle\). The paper characterizes cases in which \(\langle T \rangle = \text{sat}(T)\) in terms of a property of \(T\) called \textit{primitivity}. The authors also give algorithms to test for primitivity and apply their method to inclusion tests of the form \(\text{sat}(U)\subseteq \text{sat}(T)\) for regular chains \(U,T\). The runtime of their Maple implementation is assessed by means of example.
scientific article; zbMATH DE number 5989489
scientific article