@misc{Moeller1992, author = {M{\"o}ller, H. Michael}, title = {On decomposing systems of polynomial equations with finitely many solutions.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-851}, number = {SC-92-15}, year = {1992}, abstract = {This paper deals with systems of \$m\$ polynomial equations in \$n\$ unknown, which have only finitely many solutions. A method is presented which decomposes the solution set into finitely many subsets, each of them given by a system of type \begin{displaymath} f_1(x_1)=0, f_2(x_1,x_2)=0,...,f_n(x_1,...,x_n)=0. \end{displaymath} The main tools for the decomposition are from ideal theory and use symbolical manipulations. For the ideal generated by the polynomials which describe the solution set, a lexicographical Gr{\"o}bner basis is required. A particular element of this basis allows the decomposition of the solution set. A recursive application of these decomposition techniques gives finally the triangular subsystems. The algorithm gives even for non-finite solution sets often also usable decompositions. {\bf Keywords:} Algebraic variety decomposition, Gr{\"o}bner bases, systems of nonlinear equations.}, language = {en} }