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Algebraic and Semialgebraic Proofs: Methods and Paradoxes

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  • First Online:
Automated Deduction in Geometry (ADG 2000)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 2061))

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Abstract

The aim of the present paper is the following:

  1. —

    Examine critically some features of the usual algebraic proof protocols, in particular the “test phase” that checks if a theorem is “true” or not, depending on the existence of a non-degenerate component on which it is true; this form of “truth” leads to paradoxes, that are analyzed both for real and complex theorems.

  2. —

    Generalize these proof tools to theorems on the real field; the generalization relies on the construction of the real radical, and allows to consider inequalities in the statements.

  3. —

    Describe a tool that can be used to transform an algebraic proof valid for the complex field into a proof valid for the real field.

  4. —

    Describe a protocol, valid for both complex and real theorems, in which a statement is supplemented by an example; this protocol allows us to avoid most of the paradoxes.

Work partly performed within the Project “Tecniche per Immagini”, cluster C15, progetto n. 7 “Sviluppo, Analisi ed Implementazione di Metodi Matematici Avanzati per il Trattamento di Immagini”, with the contribution of MURST.

Acknowledgments

We want to thank Fabrizio Broglia for useful discussions and references on the semialgebraic geometry issues discussed in this paper.

We thank the anonymous referees for many useful remarks.

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Authors and Affiliations

  1. Dipartimento di Matematica, Via Buonarroti 2, I-56127, Pisa, Italy

    Pasqualina Conti & Carlo Traverso

Authors
  1. Pasqualina Conti
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  2. Carlo Traverso
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Editors and Affiliations

  1. Zentrum Mathematik, SB4, Technische Universität München, 80290, München, Germany

    Jürgen Richter-Gebert

  2. Laboratoire d’Informatique de Paris 6, Université Pierre et Marie Curie — CNRS, 4 place Jussieu, 75252, Paris Cedex 05, France

    Dongming Wang

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Conti, P., Traverso, C. (2001). Algebraic and Semialgebraic Proofs: Methods and Paradoxes. In: Richter-Gebert, J., Wang, D. (eds) Automated Deduction in Geometry. ADG 2000. Lecture Notes in Computer Science(), vol 2061. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45410-1_6

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  • DOI: https://doi.org/10.1007/3-540-45410-1_6

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  • Print ISBN: 978-3-540-42598-4

  • Online ISBN: 978-3-540-45410-6

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