Algorithmic \(P\)- distributions and applications (Q1200970)

Goal of this paper is to represent a given (projective) algebraic variety in \(\mathbb{P}^ n\) as set theoretical complete intersection of hypersurfaces of minimal number and to realize this representation on computers. Given an ideal \(I\subset R=K[X_ 1,\dots,X_ n]\), \(K\) a field, one has to find an algorithm which gives homogeneous polynomials \(f_ 1,\dots,f_ r\) of minimal number such that rad\((f_ 1,\dots,f_ r)R=\text{rad} I\). As solution of this problem the authors find the so called algorithmic \(P\)-decomposition and they show by examples possibilities of its application to computers. Algorithmic \(P\)- decompositions are special \(P\)-decompositions with the following definition: Let \(R\) be a commutative ring with unit, let \(P\subset R\) be finite and \(P_ i\subset P\) subsets, \(i=0,\dots,r\). \(\{P_ 0,\dots,P_ r\}\) is named `\(P\)-decomposition', if the following holds: (1) \(\bigcup^ r_{i=0}P_ i=P\); (2) \(P_ 0\) consists of one element; (3) For \(p,p''\in P_ i\), \(0<i\leq r\), \(p\neq p'\), one can find \(j\), \(0\leq j<i\), and \(p'\in P_ i\), such that \(p\cdot p''\in p'R\).