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SC 92-15

H. Michael Möller On decomposing systems of polynomial equations with finitely many solutions. Appeared in: Applicable Algebra in Engineering, Communication and Computing (AAECC), 4 (1993) pp. 217-230

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

$$f_1(x_1)=0, f_2(x_1,x_2)=0,...,f_n(x_1,...,x_n)=0.$$

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 Gröbner 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.
Keywords: Algebraic variety decomposition, Gröbner bases, systems of nonlinear equations.
Keywords: Algebraic variety decomposition, Groebner bases, systems of nonlinear equations