### Abstract

The derivation of a Hardy field induces on its value group a certain function ψ. If a Hardy field extends the real field and is closed under powers, then its value group is also a vector space over R. Such "ordered vector spaces with ψ-function" are called H-couples. We define closed H-couples and show that every H-couple can be embedded into a closed one. The key fact is that closed H-couples have an elimination theory: solvability of an arbitrary system of equations and inequalities (built up from vector space operations, the function ψ, parameters, and the unknowns to be solved for) is equivalent to an effective condition on the parameters of the system. The H-couple of a maximal Hardy field is closed, and this is also the case for the H-couple of the field of logarithmic-exponential series over R. We analyze in detail finitely generated extensions of a given H-couple.

Original language | English (US) |
---|---|

Pages (from-to) | 309-358 |

Number of pages | 50 |

Journal | Journal of Algebra |

Volume | 225 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2000 |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Algebra*,

*225*(1), 309-358. https://doi.org/10.1006/jabr.1999.8128

**Closed Asymptotic Couples.** / Aschenbrenner, Matthias; Van Den Dries, Lou.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 225, no. 1, pp. 309-358. https://doi.org/10.1006/jabr.1999.8128

}

TY - JOUR

T1 - Closed Asymptotic Couples

AU - Aschenbrenner, Matthias

AU - Van Den Dries, Lou

PY - 2000/1/1

Y1 - 2000/1/1

N2 - The derivation of a Hardy field induces on its value group a certain function ψ. If a Hardy field extends the real field and is closed under powers, then its value group is also a vector space over R. Such "ordered vector spaces with ψ-function" are called H-couples. We define closed H-couples and show that every H-couple can be embedded into a closed one. The key fact is that closed H-couples have an elimination theory: solvability of an arbitrary system of equations and inequalities (built up from vector space operations, the function ψ, parameters, and the unknowns to be solved for) is equivalent to an effective condition on the parameters of the system. The H-couple of a maximal Hardy field is closed, and this is also the case for the H-couple of the field of logarithmic-exponential series over R. We analyze in detail finitely generated extensions of a given H-couple.

AB - The derivation of a Hardy field induces on its value group a certain function ψ. If a Hardy field extends the real field and is closed under powers, then its value group is also a vector space over R. Such "ordered vector spaces with ψ-function" are called H-couples. We define closed H-couples and show that every H-couple can be embedded into a closed one. The key fact is that closed H-couples have an elimination theory: solvability of an arbitrary system of equations and inequalities (built up from vector space operations, the function ψ, parameters, and the unknowns to be solved for) is equivalent to an effective condition on the parameters of the system. The H-couple of a maximal Hardy field is closed, and this is also the case for the H-couple of the field of logarithmic-exponential series over R. We analyze in detail finitely generated extensions of a given H-couple.

UR - http://www.scopus.com/inward/record.url?scp=0034148623&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034148623&partnerID=8YFLogxK

U2 - 10.1006/jabr.1999.8128

DO - 10.1006/jabr.1999.8128

M3 - Article

AN - SCOPUS:0034148623

VL - 225

SP - 309

EP - 358

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 1

ER -