ON DECOMPOSING SYSTEMS OF POLYNOMIAL EQUATIONS WITH FINITELY MANY SOLUTIONS

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 f1(x1)=0, f2(x1,x2)=0, ..., f(n)(x1, ..., x(n))=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 Grobner 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.