On decomposing systems of polynomial equations with finitely many solutions (Q1311615)

Problem: Find the common zeros of a multivariate polynomial system \(\{f_ 1,\dots,f_ k\}\). Starting point is a Gröbner basis \(G\) under lexicographical term order for the ideal \(A=(f_ 1,\dots,f_ k)\). The ideal is then decomposed into components where each lexicographical Gröbner basis has a pure triangular form. In contrast to other methods the algorithm is based only on ideal quotients \(I:f^ m\) and ideal sums \(I+(f)\) which can be computed using the Buchberger algorithm. Neither factorization nor full primary decomposition are needed. A proof for the method is given for the case of \(\dim=0\). However, the method can be useful also for many cases of higher dimension.