OAL-RAG 2024: Tomás Recio (Universidad Antonio de Nebrija)

OAL-RAG 2024: Tomás Recio (Universidad Antonio de Nebrija)

by Tomás Recio and M. Pilar Vélez

Formal & Physical Sciences Mathematics PBMGeometryPBMWAlgebraic geometry
🎙 Tomás Recio and M. Pilar Vélez 👥 498 📅 July 7, 2026 ⏱ 46 min 👁 4 📄 expert opinion 🧭 2026-08-16
Available in: English (current) Français

Keywords

GeoGebra Discoveryautomated theorem provingreal algebraic geometryGröbner basiscomplexity measure

Summary

The talk by Tomás Recio and M. Pilar Vélez, presented at OAL-RAG 2024, focuses on the development and application of GeoGebra Discovery, a prototype that extends GeoGebra with automated reasoning tools for geometry. The speakers begin by recalling their collaboration since 1996 and the evolution of GeoGebra into a dynamic mathematics system with symbolic computation capabilities. They then demonstrate GeoGebra Discovery’s features, including commands like ‘relation’, ‘prove’, and ‘soproof’, which allow users to discover and prove geometric theorems automatically. The underlying algorithms rely on computational algebraic geometry, primarily over the complex numbers, using Gröbner bases and elimination ideals. The talk also addresses the extension to real algebraic geometry, handling inequalities via quantifier elimination. A significant part of the presentation introduces the ‘automatic geometer’, a tool that automatically generates theorems from a given configuration, and proposes a complexity measure to distinguish trivial from interesting results. The speakers discuss challenges in extending these methods to the real case, such as computing certificates for real radical membership and handling inequalities. They conclude by inviting collaboration on these open problems.

176 words

Critical Evaluation

Value of the Information & Strength of the Argument

The talk provides valuable insights into the state-of-the-art of automated reasoning in geometry, showcasing practical tools and concrete examples. The argumentation is solid, grounded in the speakers’ extensive experience and the demonstrated performance of GeoGebra Discovery. The proposed complexity measure is innovative and well-illustrated with examples from Pythagoras to an olympiad problem, suggesting its potential utility. The discussion of open problems, particularly in the real algebraic geometry context, highlights the limitations and future directions of the field.

Scientific Rigor, Source Quality, Title Accuracy

The presentation is scientifically rigorous, referencing specific software (GeoGebra, GeoGebra Discovery, Tarski) and mathematical concepts (Gröbner bases, real radical). The sources cited are primarily the software and the speakers’ own work, which are appropriate for a talk of this nature. The title accurately reflects the content, focusing on the speaker and the topic. No comments were provided for analysis.

151 words

Title / Content Match

The title accurately reflects the speaker and the topic, focusing on the use of GeoGebra Discovery for automated reasoning in geometry.

Quality & Reliability

8/10

The talk is given by established researchers in the field, presenting their own work and referencing concrete software and algorithms. The content is technical and appears accurate, though it is a presentation of ongoing research rather than a peer-reviewed publication.

Key Moments

Markers derived by PSI from the transcript: the creator did not define chapters.

Cited Sources

  • GeoGebra — Mentioned as the dynamic mathematics system used as the basis for GeoGebra Discovery.
  • GeoGebra Discovery — The prototype software developed by the speakers for automated reasoning in geometry.
  • Tarski — Mentioned as a tool for real quantifier elimination used in GeoGebra Discovery.
  • Giac — Mentioned as a computer algebra system used for algebraic geometry computations.

Concurring Sources

  • GeoGebra Discovery — The software's official page, which includes documentation and examples.
  • Gröbner basis — Wikipedia article explaining the concept used in the algorithms.

Contribution & Novelties

The talk presents GeoGebra Discovery as an accessible tool for automated theorem proving in geometry, integrating complex and real algebraic geometry methods. The novel contribution is the proposal of a complexity measure for geometric theorems, based on the degree of polynomials needed to express the thesis as a combination of hypotheses. This measure could help filter trivial from interesting results in automated theorem generation. The talk also highlights open problems in extending these methods to inequalities and real radical membership.

Pour aller plus loin :

  • GeoGebra Discovery — The software discussed, with online and desktop versions.
  • Gröbner basis — The algebraic tool underlying the algorithms.
  • Real algebraic geometry — The field addressing real solutions to polynomial equations and inequalities.
  • Tarski–Seidenberg theorem — The theoretical basis for quantifier elimination over the reals.

131 words

Radar Profile

The radar profile shows high scores in quantity and quality of information, technical level, and global reliability, indicating a dense and expert-level presentation. The lower score in adequacy of title suggests that while the title is accurate, it may not fully capture the depth of the content.

Reliability 8/10