### Abstract

We introduce new algorithms for quantifier eliminations (QE) in the domain of an algebraically closed field. Our algorithms are based on the computation of comprehensive Gröbner systems (CGS). We study Suzuki–Sato’s CGS computation algorithm and its successors in more detail and modify them into an optimal form for applying to QE. Based on this modified algorithm, we introduce two QE algorithms. One is pursuing the simplest output quantifier free formula which employs only the computations of CGS. The other consists of parallel computations of CGS and GCD of parametric unary polynomials. It achieves faster computation time with reasonably simple output quantifier free formulas. Our implementation shows that in many examples our algorithms are superior to other existing algorithms such as a QE algorithm adopted in the Mathematica package Reduce and Resolve or a QE algorithm adopted in the Maple package Projection which are the most efficient existing implementations as far as we know.

Original language | English |
---|---|

Pages (from-to) | 267-281 |

Number of pages | 15 |

Journal | Mathematics in Computer Science |

Volume | 9 |

Issue number | 3 |

DOIs | |

Publication status | Published - Oct 1 2015 |

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### All Science Journal Classification (ASJC) codes

- Computational Mathematics
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*Mathematics in Computer Science*,

*9*(3), 267-281. https://doi.org/10.1007/s11786-015-0237-x

**On QE Algorithms over an Algebraically Closed Field Based on Comprehensive Gröbner Systems.** / Fukasaku, Ryoya; Inoue, Shutaro; Sato, Yosuke.

Research output: Contribution to journal › Article

*Mathematics in Computer Science*, vol. 9, no. 3, pp. 267-281. https://doi.org/10.1007/s11786-015-0237-x

}

TY - JOUR

T1 - On QE Algorithms over an Algebraically Closed Field Based on Comprehensive Gröbner Systems

AU - Fukasaku, Ryoya

AU - Inoue, Shutaro

AU - Sato, Yosuke

PY - 2015/10/1

Y1 - 2015/10/1

N2 - We introduce new algorithms for quantifier eliminations (QE) in the domain of an algebraically closed field. Our algorithms are based on the computation of comprehensive Gröbner systems (CGS). We study Suzuki–Sato’s CGS computation algorithm and its successors in more detail and modify them into an optimal form for applying to QE. Based on this modified algorithm, we introduce two QE algorithms. One is pursuing the simplest output quantifier free formula which employs only the computations of CGS. The other consists of parallel computations of CGS and GCD of parametric unary polynomials. It achieves faster computation time with reasonably simple output quantifier free formulas. Our implementation shows that in many examples our algorithms are superior to other existing algorithms such as a QE algorithm adopted in the Mathematica package Reduce and Resolve or a QE algorithm adopted in the Maple package Projection which are the most efficient existing implementations as far as we know.

AB - We introduce new algorithms for quantifier eliminations (QE) in the domain of an algebraically closed field. Our algorithms are based on the computation of comprehensive Gröbner systems (CGS). We study Suzuki–Sato’s CGS computation algorithm and its successors in more detail and modify them into an optimal form for applying to QE. Based on this modified algorithm, we introduce two QE algorithms. One is pursuing the simplest output quantifier free formula which employs only the computations of CGS. The other consists of parallel computations of CGS and GCD of parametric unary polynomials. It achieves faster computation time with reasonably simple output quantifier free formulas. Our implementation shows that in many examples our algorithms are superior to other existing algorithms such as a QE algorithm adopted in the Mathematica package Reduce and Resolve or a QE algorithm adopted in the Maple package Projection which are the most efficient existing implementations as far as we know.

UR - http://www.scopus.com/inward/record.url?scp=84943271675&partnerID=8YFLogxK

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U2 - 10.1007/s11786-015-0237-x

DO - 10.1007/s11786-015-0237-x

M3 - Article

AN - SCOPUS:84943271675

VL - 9

SP - 267

EP - 281

JO - Mathematics in Computer Science

JF - Mathematics in Computer Science

SN - 1661-8270

IS - 3

ER -