{"id":{"repo_id":"dialnet","oai_identifier":"oai:dialnet.unirioja.es:TES0000023233"},"canonical_url":"https://search.dev.ndltd.org/etd/dialnet/oai:dialnet.unirioja.es:TES0000023233","repository":{"repo_id":"dialnet","name":"Dialnet","base_url":"https://dialnet.unirioja.es/oaites/OAIHandler"},"display":{"title":"Kaleidoscopic decomposition, classification of finite branched coverings of the sphere and path-lifting algorithms","abstract":"The aim of this work is two-fold: On the one hand, to study the finite branched coverings of the Riemann sphere, providing original contributions with regard to their structure and properties, with a particular focus on rational maps and polynomials. On the other hand, to employ the introduced theory and original results of this memoir to design and implement novel numerical path-lifting procedures. In order to meet these purposes, we study finite branched coverings between compact Riemann surfaces, introducing an original canonical decomposition, which we call kaleidoscopic decomposition, that will be employed throughout the memoir to present a novel analysis of different topological aspects as the monodromy action or the regularity. Moreover, we study rational maps and polynomials as branched self-coverings of the Riemann sphere by considering an original approach based on finite combinatorial groupoids, which allows us to prove novel results addressing some of their topological properties (like the monodromy or the group of automorphisms), to completely characterize their regularity in simple algebraic terms, and to study and relate different notions of classification for these families of maps. As a consequence of the aforementioned study, we design novel algorithms supported by the path-lifting properties that rational and polynomial maps exhibit, which are devoted to different purposes, like the approximation of roots, the computation of the monodromy action, or the description of the group of automorphisms. The implementation of such algorithms is tested through two different cases of study, yielding an exploration of some numerical aspects of the considered path-lifting methods.","abstract_html":"The aim of this work is two-fold: On the one hand, to study the finite branched coverings of the Riemann sphere, providing original contributions with regard to their structure and properties, with a particular focus on rational maps and polynomials. On the other hand, to employ the introduced theory and original results of this memoir to design and implement novel numerical path-lifting procedures. In order to meet these purposes, we study finite branched coverings between compact Riemann surfaces, introducing an original canonical decomposition, which we call kaleidoscopic decomposition, that will be employed throughout the memoir to present a novel analysis of different topological aspects as the monodromy action or the regularity. Moreover, we study rational maps and polynomials as branched self-coverings of the Riemann sphere by considering an original approach based on finite combinatorial groupoids, which allows us to prove novel results addressing some of their topological properties (like the monodromy or the group of automorphisms), to completely characterize their regularity in simple algebraic terms, and to study and relate different notions of classification for these families of maps. As a consequence of the aforementioned study, we design novel algorithms supported by the path-lifting properties that rational and polynomial maps exhibit, which are devoted to different purposes, like the approximation of roots, the computation of the monodromy action, or the description of the group of automorphisms. The implementation of such algorithms is tested through two different cases of study, yielding an exploration of some numerical aspects of the considered path-lifting methods.","abstract_has_math":false,"creators":["Álvarez Aparicio, Víctor"],"institution":"Universidad de La Rioja (España)","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Gutiérrez Jiménez, José Manuel (null)","Hernández Paricio, Luis Javier (null)","Rivas Rodríguez, María Teresa (null)"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2026,"date_issued":"2026","date_published":"2026","updated_at":"2026-07-24T06:27:11Z","subjects":[],"languages":["eng"],"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"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://dialnet.unirioja.es/servlet/oaites?codigo=399608","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Gutiérrez Jiménez, José Manuel (null)","Hernández Paricio, Luis Javier (null)","Rivas Rodríguez, María Teresa (null)"]},{"key":"dc:creator","label":"Author","values":["Álvarez Aparicio, Víctor"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2026"]},{"key":"dc:publisher","label":"Institution","values":["Universidad de La Rioja (España)"]},{"key":"dc:type","label":"Dc Type","values":["text (thesis)"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["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"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://dialnet.unirioja.es/servlet/oaites?codigo=399608"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The aim of this work is two-fold: On the one hand, to study the finite branched coverings of the Riemann sphere, providing original contributions with regard to their structure and properties, with a particular focus on rational maps and polynomials. On the other hand, to employ the introduced theory and original results of this memoir to design and implement novel numerical path-lifting procedures. In order to meet these purposes, we study finite branched coverings between compact Riemann surfaces, introducing an original canonical decomposition, which we call kaleidoscopic decomposition, that will be employed throughout the memoir to present a novel analysis of different topological aspects as the monodromy action or the regularity. Moreover, we study rational maps and polynomials as branched self-coverings of the Riemann sphere by considering an original approach based on finite combinatorial groupoids, which allows us to prove novel results addressing some of their topological properties (like the monodromy or the group of automorphisms), to completely characterize their regularity in simple algebraic terms, and to study and relate different notions of classification for these families of maps. As a consequence of the aforementioned study, we design novel algorithms supported by the path-lifting properties that rational and polynomial maps exhibit, which are devoted to different purposes, like the approximation of roots, the computation of the monodromy action, or the description of the group of automorphisms. The implementation of such algorithms is tested through two different cases of study, yielding an exploration of some numerical aspects of the considered path-lifting methods.","Este trabajo tiene un propósito doble: Por un lado, estudiar las cubiertas ramificadas finitas de la esfera de Riemann, presentando aportaciones originales con respecto a su estructura y propiedades, con un interés particular en las funciones racionales y los polinomios. Por otro lado, emplear los resultados novedosos introducidos en esta memoria para diseñar e implementar nuevos algoritmos de levantamiento de caminos de carácter numérico. Para atender a estos propósitos, estudiaremos las cubiertas ramificadas finitas entre superficies de Riemann compactas, introduciendo una descomposición canónica original de este trabajo, llamada descomposición caleidos\\-cópica, que será empleada a lo largo de la memoria para analizar algunos de sus aspectos topológicos, como la monodromía o la regularidad. Además, estudiaremos las funciones racionales y los polinomios como cubiertas ramificadas de la esfera de Riemann adoptando un enfoque novedoso basado en grupoides combinatorios finitos, que permite probar nuevos resultados referentes a algunas de sus propiedades topológicas (como la monodromía o el grupo de automorfismos), caracterizar su regularidad en términos algebraicos sencillos, y analizar y relacionar distintas nociones de clasificación. Como consecuencia de dicho estudio, diseñaremos nuevos algoritmos basados en propiedades de levantamiento de caminos de las funciones racionales y polinomios, con diversos propósitos como la aproximación de raíces, el cálculo de la acción de monodromía, o la descripción del grupo de automorfismos. La implementación de estos algoritmos será estudiada a través de dos casos de uso, cuyos resultados motivarán la exploración de algunos aspectos numéricos de los métodos de levantamiento de caminos considerados."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Kaleidoscopic decomposition, classification of finite branched coverings of the sphere and path-lifting algorithms"]}]}],"canonical_facts":{"dc:contributor":["Gutiérrez Jiménez, José Manuel (null)","Hernández Paricio, Luis Javier (null)","Rivas Rodríguez, María Teresa (null)"],"dc:creator":["Álvarez Aparicio, Víctor"],"dc:date":["2026"],"dc:description":["The aim of this work is two-fold: On the one hand, to study the finite branched coverings of the Riemann sphere, providing original contributions with regard to their structure and properties, with a particular focus on rational maps and polynomials. On the other hand, to employ the introduced theory and original results of this memoir to design and implement novel numerical path-lifting procedures. In order to meet these purposes, we study finite branched coverings between compact Riemann surfaces, introducing an original canonical decomposition, which we call kaleidoscopic decomposition, that will be employed throughout the memoir to present a novel analysis of different topological aspects as the monodromy action or the regularity. Moreover, we study rational maps and polynomials as branched self-coverings of the Riemann sphere by considering an original approach based on finite combinatorial groupoids, which allows us to prove novel results addressing some of their topological properties (like the monodromy or the group of automorphisms), to completely characterize their regularity in simple algebraic terms, and to study and relate different notions of classification for these families of maps. As a consequence of the aforementioned study, we design novel algorithms supported by the path-lifting properties that rational and polynomial maps exhibit, which are devoted to different purposes, like the approximation of roots, the computation of the monodromy action, or the description of the group of automorphisms. The implementation of such algorithms is tested through two different cases of study, yielding an exploration of some numerical aspects of the considered path-lifting methods.","Este trabajo tiene un propósito doble: Por un lado, estudiar las cubiertas ramificadas finitas de la esfera de Riemann, presentando aportaciones originales con respecto a su estructura y propiedades, con un interés particular en las funciones racionales y los polinomios. Por otro lado, emplear los resultados novedosos introducidos en esta memoria para diseñar e implementar nuevos algoritmos de levantamiento de caminos de carácter numérico. Para atender a estos propósitos, estudiaremos las cubiertas ramificadas finitas entre superficies de Riemann compactas, introduciendo una descomposición canónica original de este trabajo, llamada descomposición caleidos\\-cópica, que será empleada a lo largo de la memoria para analizar algunos de sus aspectos topológicos, como la monodromía o la regularidad. Además, estudiaremos las funciones racionales y los polinomios como cubiertas ramificadas de la esfera de Riemann adoptando un enfoque novedoso basado en grupoides combinatorios finitos, que permite probar nuevos resultados referentes a algunas de sus propiedades topológicas (como la monodromía o el grupo de automorfismos), caracterizar su regularidad en términos algebraicos sencillos, y analizar y relacionar distintas nociones de clasificación. Como consecuencia de dicho estudio, diseñaremos nuevos algoritmos basados en propiedades de levantamiento de caminos de las funciones racionales y polinomios, con diversos propósitos como la aproximación de raíces, el cálculo de la acción de monodromía, o la descripción del grupo de automorfismos. La implementación de estos algoritmos será estudiada a través de dos casos de uso, cuyos resultados motivarán la exploración de algunos aspectos numéricos de los métodos de levantamiento de caminos considerados."],"dc:format":["application/pdf"],"dc:identifier":["https://dialnet.unirioja.es/servlet/oaites?codigo=399608"],"dc:language":["eng"],"dc:publisher":["Universidad de La Rioja (España)"],"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"],"dc:title":["Kaleidoscopic decomposition, classification of finite branched coverings of the sphere and path-lifting algorithms"],"dc:type":["text (thesis)"]},"updated_at":"2026-07-24T06:27:11Z"}