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By H. Werner, H.J. Bünger

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Geddes, S. Czapor, and G. Labahn. Algorithms for Computer Algebra. Kluwer Academic Publisher, 1992. [7] D. Kunth. The Art of Computer Programming. Vol. II, Addison-Wesley, 1981. [8] F. Winkler. Polynomial Algorithms in Computer Algebra. Springer, 1996. [9] M. Monagan. Maximal quotient rational reconstruction: an almost optimal algorithm for rational reconstruction. In the Proceedings of ISSAC 2004, pp. 243-249. S. T. H. Davenport. p-adic Reconstruction of rational numbers. SIGSAM Bulletin, 16, No.

J. Encarnacion. Efficient rational number reconstruction. J. of Symbolic Computation 20, pp. 299–297, 1995. [3] G. Collins, R. Loos, and F. Winkler. Arithmetic in basic algebraic domains. In [1], pages 189–220. [4] R. Feng, X. Gao. Rational general solutions of algebraic ordinary differential equations. In the Proceedings of ISSAC 2004, pp. 155-161. [5] J. von zur Gathen, J. Gerhard. Modern Computer Algebra, Cambridge Press, First Edition, 1999. 30 [6] K. Geddes, S. Czapor, and G. Labahn. Algorithms for Computer Algebra.

The next theorem illustrates a geometric interpretation of saturation of ideals. Theorem 9 Let I be an ideal of R and g in R. Let ¯ n |ξ ∈ V(I) and g(ξ) = 0}. S = {ξ ∈ K Let J = I : g ∞ . Then V(J) is the smallest algebraic set containing S. Proof. It is straightforward to show that S ⊂ V(J). Let H be an ideal of R such that S ⊂ V(H). We √ to prove that V(J) ⊂ V(H). By Theorem 8 √ have we need only to show H ⊂ J. Assume that h ∈ (H), then h vanishes on S. Hence gh vanishes on V(I). Theorem 8 √ then implies that (gh)k ∈ I k for some k ∈ N.

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