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The dimension method in elementary and differential geometry

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Abstract

The paper discusses the dimension method, introduced by Carrà and Gallo, to prove theorems in elementary and differential geometry. The method, based on the computation of the dimension of algebraic varieties, which is carried out by first determining Gröbner bases, introduces different levels of validity of geometric statements. The paper compares the outcome of this approach with that of Wu's method, discusses its theoretical bases and presents a refined decision algorithm for the differential case, which improves a previous version. A software environment that allows to experiment with the two different theorem provers has been designed and is presented in the paper. The environment interface provides a specification language that enables users to easily describe geometric relationships among the geometric entities involved in a problem. A problem description is then interpreted to yield a graphic representation of the problem and to provide polynomial equations which constitute the input to the provers.

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Authors and Affiliations

  1. Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, Via Comelico 39, I-20135, Milano, Italy

    M. A. Alberti, B. Lammoglia & M. Torelli

  2. Dipartimento di Matematica, Università di Catania, Viale A. Doria 6, I-95125, Catania, Italy

    G. Carrà Ferro

Authors
  1. M. A. Alberti
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  2. G. Carrà Ferro
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  3. B. Lammoglia
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  4. M. Torelli
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Additional information

This paper is based on two different talks given at the workshop “Algebraic Approaches to Geometric Reasonin”, Linz, Austria, August 1992. The work has been supported by the Italian National Project “Calcolo algebrico e simbolico”, Ministero dell'Università e della Ricerca Scientifica e Tecnologica.

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Alberti, M.A., Carrà Ferro, G., Lammoglia, B. et al. The dimension method in elementary and differential geometry. Ann Math Artif Intell 13, 47–71 (1995). https://doi.org/10.1007/BF01531323

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