On decomposing systems of polynomial equations with finitely many solutions

Abstract

This paper deals with systems ofm polynomial equations inn 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

$$f_1 \left( {x_1 } \right) = 0,f_2 \left( {x_1 ,x_2 } \right) = 0, \ldots ,f_n \left( {x_1 , \ldots ,x_n } \right) = 0$$

. 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 Gröbner basis is required. A particular element of this basis allows the decomposition of the solution set. By a recursive application of these decomposition techniques the triangular subsystems are finally obtained. The algorithm gives even for non-finite solution sets often also usable decompositions.