Localization and primary decomposition of polynomial ideals
The authors give a new algorithm for primary decomposition of a polynomial ideal. Let \(I\) be an ideal of the polynomial ring \(R=\mathbb{Q}[x_1,\dots,x_n]\) over the rational numbers. An ideal is called pseudo-primary, if its radical is a prime ideal. The methods are described roughly as follows. First compute a pseudo-primary decomposition \(I=\overline{Q}_1\cap\dots\cap \overline{Q}_r \cap I'\), where \(\overline{Q}_1,\dots,\overline{Q}_r\) are pseudo-primary ideals and either \(I'=R\) or \(\dim(I')<\dim(I)\). This is done using the prime decomposition of the radical of \(I\) and a system of separators. Next, for each \(\overline{Q}_i\) compute its extraction \(\overline{Q}_i=Q_i\cap I_i'\), where \(Q_i\) is a unique isolated primary component of \(\overline{Q}_i\) and either \(I_i'=R\) or \(\dim(I_i') < \dim(\overline{Q}_i)\). Add this \(Q_i\) to \(\mathcal Q\) being constructed. If \(I'\neq R\) or \(I_i'\neq R\), then recursively apply this procedure to \(I'\) or \(I_i'\). Thus, we get a general primary decomposition \(\mathcal Q\) of \(I\). Finally eliminate redundant components from \(\mathcal Q\) to get a shortest irredundant decomposition. The authors give a method to avoid unnecessary recursive calls that give redundant primary components. They also give experimental results comparing their methods with other existing methods.
primary decomposition of a polynomial ideal
scientific article; zbMATH DE number 987581