Leilei Guo
ABSTRACT
Based on Eisenbud's idea (see [Eisenbud, D., Evans, G., 1973. Every algebraic set in n-space is the intersection of n hypersurfaces. Invent. Math. 19, 107-112]), we present an algorithm for computing set-theoretic generators for any algebraic variety in the affine n-space, which consists of at most n polynomials. With minor modifications, this algorithm is also valid for projective algebraic variety in projective n-space.
- Abhyankar, S.S., 1973. On Macaulay's example. In: Conf. Comm. Algebra, Lawrence 1972, Springer LNM, vol311, 1--16.Google Scholar
Cross Ref
- Atiyah, M.F., Macdonald, I.G., 1969. Introduction to Commutative Algebra, Addison-Wesley Publishing Company.Google Scholar
- Chou, S.C. and Gao, X.S., 1991. On the dimension of an arbitrary ascending chain. Chinese Bull. of Sci. 38, 799--804.Google Scholar
- Cox, D., Little, J., O'Shea, D., 1996. Ideals, Varieties, and Algorithms, Second Edition. Springer.Google Scholar
- Eisenbud, D., Evans, G., 1973. Every algebraic set in n-space is the intersection of n hypersurfaces. Invent. Math. 19, 107--112.Google ScholarCross Ref
- Eisenbud, D., Sturmfels, B., 1994. Finding sparse systems of parameters. Journal of Pure and Applied Algebra, 143--157.Google ScholarCross Ref
- Eisenbud, D., 1994. Commutative Algebra with a View toward Algebraic Geometry. Graduate Texts in Mathematics. Springer-Verlag New York.Google Scholar
- Fortuna, E., Gianni, P., Trager, B., 2009. Generators of the ideal of an algebraic space curve. Journal of Symbolic Computation 44, 1234--1254. Google ScholarDigital Library
- Gianni, P., Trager, B., Zacharias, G., 1988. Grobner bases and primary decomposition of polynomial ideals. Journal of Symbolic Computation 6, 149--167. Google ScholarDigital Library
- Hubert, E., 2003. Notes on triangular sets and triangulation-decomposition algorithms I: Polynomial systems. In Symbolic and Numerical Scientific Computing 2001, pages 1--39. Google ScholarDigital Library
- Jia, X., Wang, H., Goldmand, R., 2010. Set-theoretic generators of rational space curves. Journal of Symbolic Computation 45, 414--433. Google ScholarDigital Library
- Kneser, M., 1960. Uber die Darstellung algebraischer Raumkurven als Durchschnitte von Flachen. Arch. Math. 11, 157--158.Google ScholarCross Ref
- Kalkbrener, M., 1993. A generalized euclidean algorithm for computing triangular representations of algebraic varieties. J. Symb. Comp.15, 143--167. Google ScholarDigital Library
- Kronecher, L., 1882. Grundzuge einer arithmetischen Theorie der algebraischen Großen. J. Reine Angew. Math. 92, 1--123.Google Scholar
- Lemaire, F., Moreno Maza, M., Pan, W., and Xie, Y., 2008. When does (T) equal Sat(T)?. In Proc. ISSAC'2008, 207--214. ACM Press. Google ScholarDigital Library
- Laplagne, S., 2006. An algorithm for the computation of the radical of an ideal. Proceedings of the 2006 international symposium on Symbolic and algebraic computation, July 09--12, Genoa, Italy. Google ScholarDigital Library
- Maza, M.M., Rioboo, R., 1995. Polynomial gcd computations over towers of algebraic extensions. In Applied algebra, algebraic algorithms and errorcorrecting codes, 365--382. Springer, Berlin. Google ScholarDigital Library
- Moh, T.T., 1974. On the unboundedness of generators of prime ideals in power series rings of three variables, J.Math.Soc.Japan 26, 722--734.Google ScholarCross Ref
- Sturmfels, B., 2002. Solving Systems of Polynomial Equations. Amer.Math.Soc.Google Scholar
- Schauenburg, P., 2007. A Grobner-based treatment of elimination theory for affine varieties. Journal of Symbolic Computation 42, 859--870. Google ScholarDigital Library
- Van der Waerden, B.L., 1941. Review. Zentralblatt fur Math. 24, 276.Google Scholar
- Wu, W.T., 2003. Mathematics Machenization. Science Press/Kluwer, Beijing.Google Scholar
- Yang, L. and Zhang, J., 1991. Searching dependency between algebraic equations: an algorithm applied to automated reasoning. Technical report ic/91/6, International Atomic Engery Angency, Miramare, Trieste, Italy.Google Scholar
Index Terms
- An algorithm for computing set-theoretic generators of an algebraic variety
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