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**On the computational complexity and geometry of the first-order theory of the reals. III: Quantifier elimination.**
*(English)*
Zbl 0798.68073

Summary: This series of papers [part I, ibid. 13, No. 3, 255-299 (1992; Zbl 0763.68042); part II, ibid. 13, No. 3, 301-327 (1992; Zbl 0763.68043)] presents a complete development and complexity analysis of a decision method, and a quantifier elimination method, for the first order theory of the reals. The complexity upper bound which are established are the best presently available, and both for sequential and parallel computation, and both for the bit model of computation and the real number model of computation; except for the bounds pertaining to the sequential decision method in the bit model of computation, all bounds represent significant improvements over previously established bounds.

### MSC:

68Q25 | Analysis of algorithms and problem complexity |

14P10 | Semialgebraic sets and related spaces |

03C10 | Quantifier elimination, model completeness, and related topics |

03B25 | Decidability of theories and sets of sentences |

### Keywords:

complexity analysis; decision method; quantifier elimination; first order theory of the reals; parallel computation; bit model of computation; real number model of computation
Full Text:
DOI

### References:

[1] | Ben-Or, M.; Kozen, D.; Reif, J., The complexity of elementary algebra and geometry, J. Computer Syst. Sci., 32, 251-264 (1986) · Zbl 0634.03031 |

[2] | Brown, W.; Traub, J., On Euclid’s algorithm and the theory of subresultants, J. Association for Computing Machinery, 18, 505-514 (1971) · Zbl 0226.65041 |

[3] | Collins, G., Subresultants and reduced polynomial remainder sequences, J. Association for Computing Machinery, 14, 128-142 (1967) · Zbl 0152.35403 |

[4] | Csanky, L., Fast parallel matrix inversion algorithms, SIAM Journal on Computing, 5, 618-623 (1976) · Zbl 0353.68063 |

[5] | Tarski, A., A Decision Method for Elementary Algebra and Geometry (1951), University of California Press · Zbl 0044.25102 |

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