Dealing with Degeneracies in Automated Theorem Proving in Geometry

We report, through different examples, the current development in GeoGebra, a widespread Dynamic Geometry software, of geometric automated reasoning tools by means of computational algebraic geometry algorithms. Then we introduce and analyze the case of the degeneracy conditions that so often arise...

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Bibliographic Details
Published inMathematics (Basel) Vol. 9; no. 16; p. 1964
Main Authors Kovács, Zoltán, Recio, Tomas, Tabera, Luis F., Vélez, M. Pilar
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.08.2021
Subjects
Algebra
Algorithms
Altitude
automated deduction in geometry
Automated reasoning
automated reasoning in geometry
automated theorem proving in geometry
Automation
computational algebraic geometry
Construction
Deduction
Dimensional analysis
Dynamic Geometry
Food science
GeoGebra
Geometry
Hypotheses
Independent variables
Mathematics teachers
Software
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Summary:We report, through different examples, the current development in GeoGebra, a widespread Dynamic Geometry software, of geometric automated reasoning tools by means of computational algebraic geometry algorithms. Then we introduce and analyze the case of the degeneracy conditions that so often arise in the automated deduction in geometry context, proposing two different ways for dealing with them. One is working with the saturation of the hypotheses ideal with respect to the ring of geometrically independent variables, as a way to globally handle the statement over all non-degenerate components. The second is considering the reformulation of the given hypotheses ideal—considering the independent variables as invertible parameters—and developing and exploiting the specific properties of this zero-dimensional case to analyze individually the truth of the statement over the different non-degenerate components.
ISSN:2227-7390
2227-7390
DOI:10.3390/math9161964