Decomposition of polynomial sets into characteristic pairs

A characteristic pair is a pair (G, C) of polynomial sets in which G is a reduced lexicographic Gröbner basis and C is the minimal triangular set contained in G. It is said to be normal (or strong normal) if C is normal (or C is normal and its saturated ideal equals the ideal generated by G). In this paper, we show that any finite polynomial set P can be decomposed algorithmically into finitely many (strong) normal characteristic pairs with associated zero relations, which provide representations for the zero set of P in terms of those of Gröbner bases and those of triangular sets. The algorithm we propose for the decomposition makes use of the inherent connection between Ritt characteristic sets and lexicographic Gröbner bases and is based essentially on the structural properties and the computation of lexicographic Gröbner bases. Several nice properties about the decomposition and the resulting (strong) normal characteristic pairs, in particular relationships between the Gröbner basis and the triangular set in each pair, are established. Examples are given to illustrate the algorithm and some of the properties.