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Polynomial-time computation of the dimensions of components of algebraic varieties in zero-characteristic. (English) Zbl 0893.14020
An algorithm is described for the computation of the dimensions of all the irreducible components of an algebraic set over a zero characteristic ground field. The algebraic set is given as the zero set of a family of polynomials of degree $$< d$$ in $$n$$ variables and the working time of the algorithm is polynomial in the size of the input and $$d^n$$. Techniques of real algebraic geometry are used in an essential way, hence the restriction to characteristic $$0$$. In finite characteristic the best bound presently known is $$O(d^{n^2})$$, and is also due to A. L. Chistov [see J. Sov. Math. 34, 1838-1882 (1986); translation from Zap. Nauchn. Semin. Leningrad. Otd. Mat. Inst. Steklova 137, 124-188 (1984; Zbl 0561.12010)].

##### MSC:
 14P15 Real-analytic and semi-analytic sets 14Q15 Computational aspects of higher-dimensional varieties 68W30 Symbolic computation and algebraic computation
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##### References:
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