A Modified Van der Waerden Algorithm to Decompose Algebraic Varieties and Zero-Dimensional Radical Ideals
- 1k Downloads
In this paper, we introduce a modified Van der Waerden algorithm to decompose a variety into the union of irreducible varieties. We give an effective representation for irreducible varieties obtained by the algorithm, which allows us to obtain an irredundant decomposition easily. We show that in the zero dimensional case, the polynomial systems for the irreducible varieties obtained in the Van der Waerden algorithm generate prime ideals. As a consequence, we have an algorithm to decompose the radical ideal generated by a finite set of polynomials as the intersection of prime ideals and the degree of the polynomials in the computation is bounded by O(d n ) where d is the degree of the input polynomials and n is the number of variables.
KeywordsAlgebraic variety zero dimensional variety resultant irredundant decomposition
Unable to display preview. Download preview PDF.
- 1.Buchberger, B.: Gröbner bases: an algorithmic method in polynomial ideal theory. In: Bose, N.K. (ed.) Recent Trends in Multidimensional Systems theory, D.Reidel Publ. Comp. (1985)Google Scholar
- 5.Chou, S.C., Gao, X.S.: Ritt-Wu’s decomposition algorithm and geometry theorem proving. In: Stickel, M.E. (ed.) CADE 1990. LNCS, vol. 449, pp. 207–220. Springer, Heidelberg (1990)Google Scholar
- 7.Gao, X.S., Chou, S.C.: On the dimension for arbitrary ascending chains. Chinese Bull. of Scis. 38, 396–399 (1993)Google Scholar
- 13.Ritt, J.F.: Differential algebra, American Mathematical Society (1950)Google Scholar
- 14.Szántó, Á.: Computation with polynomial systems, PhD thesis, Cornell University (1999)Google Scholar
- 15.Van der Waerden, B.L.: Einfürung in die algebraischen geometrie. Springer, Berlin (1973)Google Scholar
- 16.Van der Waerden, B.L.: Modern algebra II. Frederick Ungar Pub., New York (1953)Google Scholar
- 19.Wu, W.T.: Basic principles of mechanical theorem-proving in elementary geometries. Journal of System Science and Mathematical Sciences 4, 207–235 (1984); Re-published in Journal of Automated Reasoning 2, 221–252 (1986)Google Scholar