By Otto Mutzbauer

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Gobner, "Moderne Algebraische Geometrie," Springer Verlag, Wien-Innsbruk, 1949. J. Heintz, Definabi/ity and fast quantifier elimination in algebraically closed fields, Theoret. Comput. Sci. 24 (1983), 239-277. J. Kollar, Sharp effective Nullstellensatz, J. Am. Math. Soc. 1 (1988), 963-975. A. Logar, A computational proof of the Noether's Normalization Lemma, in "Proc. AAECC-6," LN Comput. , Springer. H. Matsumura, "Commutative Algebra," Second Edition, Benjamin/Cummings, 1980. E. Mayr - A. Meyer, The complexity of the word problem for commutative semigroups and polynomial ideals, Advances in Math.

He is constructed recurrently. Let hl := h and suppose that for some k we have defined a sequence h l , ... , hTc verifying the conditions 1), 2) and 3) for k. 1), the set ofall associated prime components P of HTc := (h l , ... , hTc) such that J 'l:. P. First case: J ~ rad(h, .. · ,1m). It is easy to see that in this case P is not empty. Let PEP. 1) we deduce that P has height k. If k < m, then the complete intersection hypothesis implies that (h, ... , 1m) 'l:. P. Since h E H Tc ~ P, we see that there exists i, 1 < i $ m, verifying (lj , ...

I" . h which is a monomial in the ZH, where HE Ea(2). I1o. h together with assigned "multiplicities" lAg IAh . v2(a) - l'a(D2h), for all (h, I'h) E :F. Clearly, each 9 g(z) is a regular function times a monomial with rational exponents in the ZH , HE Ea - (Ea(l) U Ea(2»). If IA E Q, let (IA) denote the smallest integer ~ IA. Put 82 = 82(a). Write D2(z) = D21(Z) . D22(Z), where D21 (Z) is the greatest divisor of D2(Z) which is a monomial in the ZH, H E Ea - (Ea(l) U Ea(2». After a change in the coordinates i = (i, zn-d, we can assume: = = (a) where l2p(Z) = a2pl(Z)Zn-l + a2po(i) , p = 1, ...

### Analytische Geometrie by Otto Mutzbauer

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