The importance of being zero

2018 International Symposium on Symbolic and Algebraic Computation (ISSAC), July 2018, New York, NY, United StatesWe present a deterministic algorithm for deciding if a polynomial ideal, with coefficients in an algebraically closed field K of characteristic zero, of which we know just some very limited data, namely:the number n of variables, and some upper bound for the geometric degree of its zero set in Kn, is or not the zero ideal. The algorithm performs just a finite number of decisions to check whether a point is or not in the zero set of the ideal. Moreover, we extend this technique to test, in the same fashion, if the elimination of some variables in the given ideal yields or not the zero ideal. Finally, the role of this technique in the context of automated theorem proving of elementary geometry statements, is presented, with references to recent documents describing the excellent performance of the already existing prototype version, implemented in GeoGebra.Ministerio de Economía y CompetitividadEuropean Regional Development Fun

Hacia un autómata geómetra

Este artículo tiene como objetivo ilustrar al lector, muy particularmente a los colegas de la comunidad matemática española, de la RSME, de la evolución de nuestro grupo de investigación y de su trabajo a lo largo de los últimos 30 años, en pos de la creación de un autómata geómetra. Para ello, en las secciones siguientes se trazan unas pinceladas sobre la historia de las contribuciones del equipo de investigación creado en torno a los autores de este artículo, incluyendo, en particular, una descripción informal de los conceptos, problemas y métodos que fueron desarrollando. No pretende ser, no puede ser, una historia universal de la demostración automática de teoremas geométricos, pero creemos que el lector interesado encontrará datos relevantes a este respecto en las referencias que se incluyen a lo largo de este trabajo. Finalmente se presenta el resultado más «visible» de nuestro trabajo, el autómata geómetra AG, a través de diversos ejemplos, concluyendo con unas someras reflexiones acerca de su potencial impacto en el mundo educativo

The "never-proved" triangle inequality: A GeoGebra & CAS approach

We use a quite simple, yet challenging, elementary geometry statement, the so-called "never proved" (by a mathematician) theorem, introduced by Prof. Jiawei Hong in his communication to the IEEE 1986 Symposium on Foundations of Computer Science, to exemplify and analyze the current situation of achievements, ongoing improvements and limitations of GeoGebra's automated reasoning tools, as well as other computer algebra systems, in dealing with geometric inequalities. We present a large collection of facts describing the curious (and confusing) history behind the statement and its connection to automated deduction. An easy proof of the "never proved" theorem, relying on some previous (but not trivial) human work is included. Moreover, as part of our strategy to address this challenging result with automated tools, we formulate a large list of variants of the "never proved" statement (generalizations, special cases, etc.). Addressing such variants with GeoGebra Discovery, ${\texttt{Maple}}$, ${\texttt{REDUCE/Redlog}}$ or ${\texttt{Mathematica}}$ leads us to introduce and reflect on some new approaches (e.g., partial elimination of quantifiers, consideration of symmetries, relevance of discovery vs. proving, etc.) that could be relevant to consider for future improvements of automated reasoning in geometry algorithms. As a byproduct, we obtain an original result (to our knowledge) concerning the family of triangles inscribable in a given triangle