Dynamical Gröbner bases over Dedekind rings
The notion of dynamical Gröbner bases for ideals in \(R[x_1, \ldots, x_n]\), \(R\) a Dedekind ring (with zero divisors) is introduced: Let \(>\) be a monomial ordering on \(R[x_1, \ldots, x_n]\) and \(I\subset R[x_1, \ldots, x_n]\) an ideal. \(G=\{g_1, \ldots, g_l\}\subset I\) is called a special Gröbner basis for \(I\) if \(I=\langle g_1, \ldots, g_l\rangle\) and the set \(\{\text{LC}(g_1), \ldots, \text{LC}(g_l)\}\) is totally ordered under division and for each \(i\neq j\) \( \overline{S(g_i, g_j)}^G=0\). A set \(\{(S_1, G_1), \ldots, (S_k, G_k)\}\) is called a dynamical Gröbner basis for \(I\) if \(S_1, \ldots, S_k\) are finitely generated comaximal multiplicative subsets of \(R\) and in each localization \((S_i^{-1} R)[x_1, \ldots, x_n] \) \(G_i\) is a special Gröbner basis for \(I\). As an application the ideal membership problem and the computation of syzygies are treated.
scientific article; zbMATH DE number 5758104
scientific article