The Combinatorics of Involutive Bases: Theory, algorithms and applications

In this thesis, we conduct an in-depth study of monomial ideals and other related structures in commutative algebra. Our primary tool for exploring the connections between these algebraic objects and combinatorics is the use of involutive bases, along with other types of Gröbner bases that exhibit additional combinatorial properties. In particular, the nice combinatorial properties that come along with these bases enable applications in other areas, such as system reliability, topological and algebraic invariant theory, among others. Furthermore, monomial ideals have been extensively studied by numerous authors from various perspectives. One particularly significant aspect of these ideals is their role in facilitating the exchange of information between commutative algebra and combinatorics. Notably, monomial ideals---and, in some cases, certain polynomial ideals---can be associated with graphs or cellular complexes, thereby creating a "bridge" between their algebraic and combinatorial properties. In this thesis, we utilize involutive bases as a tool to establish these associations between polynomial ideals and graphs or cellular complexes, thereby constructing such bridges. Another approach to monomial ideals is through the algebraic concept of Rees algebras. The relationship between the two is an important topic in commutative algebra and algebraic geometry. The Rees algebra of a monomial ideal is an algebra with a rich combinatorial structure that encodes information about the powers of the monomial ideal as well as the syzygies of each of its powers. In fact, obtaining an explicit description of these objects is often elusive and very difficult to determine. As a result, research typically focuses on classes of ideals with a rich algebraic structure, particularly those with a well-characterized free resolution. The mentioned features make Rees algebras a natural fit for analysis using the combinatorial framework of involutive bases. In this work, we explore these objects from a new perspective, extending previous research in the area and opening the door for future advancements. Lastly, we explore some applicable results of monomial ideals and involutive bases in the field of system reliability. To improve the efficiency of our methods, we conduct a thorough performance analysis of several algorithms for computing involutive and involutive-like bases of monomial ideals. We establish a comparison between existing implemented algorithms and those we have developed, utilizing the computer algebra system CoCoALib in C++

Grantor dc:publisher

Universidad de La Rioja (España)

Year dc:date

2025

Author dc:creator

• Iglesias González, Rodrigo

Contributors dc:contributor

• Sáenz de Cabezón Irigaray, Eduardo (null)
• Seiler, Werner M. (null)

Statement dc:rights

• LICENCIA DE USO: Los documentos a texto completo incluidos en Dialnet son de acceso libre y propiedad de sus autores y/o editores. Por tanto, cualquier acto de reproducción, distribución, comunicación pública y/o transformación total o parcial requiere el consentimiento expreso y escrito de aquéllos. Cualquier enlace al texto completo de estos documentos deberá hacerse a través de la URL oficial de éstos en Dialnet. Más información: https://dialnet.unirioja.es/info/derechosOAI | INTELLECTUAL PROPERTY RIGHTS STATEMENT: Full text documents hosted by Dialnet are protected by copyright and/or related rights. This digital object is accessible without charge, but its use is subject to the licensing conditions set by its authors or editors. Unless expressly stated otherwise in the licensing conditions, you are free to linking, browsing, printing and making a copy for your own personal purposes. All other acts of reproduction and communication to the public are subject to the licensing conditions expressed by editors and authors and require consent from them. Any link to this document should be made using its official URL in Dialnet. More info: https://dialnet.unirioja.es/info/derechosOAI

Language dc:language

spa

Repository record dc:identifier

https://dialnet.unirioja.es/servlet/oaites?codigo=395305

OAI identifier oai:identifier

oai:dialnet.unirioja.es:TES0000023204