On decomposing systems of polynomial equations with finitely many solutions.
Please always quote using this URN: urn:nbn:de:0297-zib-851
- 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 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. {\bf Keywords:} Algebraic variety decomposition, Gröbner bases, systems of nonlinear equations.
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| Author: | H. Michael Möller |
|---|---|
| Document Type: | ZIB-Report |
| Tag: | Algebraic variety decomposition; Groebner bases; systems of nonlinear equations |
| Date of first Publication: | 1992/06/15 |
| Series (Serial Number): | ZIB-Report (SC-92-15) |
| ZIB-Reportnumber: | SC-92-15 |
| Published in: | Appeared in: Applicable Algebra in Engineering, Communication and Computing (AAECC), 4 (1993) pp. 217-230 |
