{"id":{"repo_id":"uwo","oai_identifier":"oai:uwo.scholaris.ca:20.500.14721/32042"},"canonical_url":"https://search.dev.ndltd.org/etd/uwo/oai:uwo.scholaris.ca:20.500.14721/32042","repository":{"repo_id":"uwo","name":"Western University","base_url":"https://uwo.scholaris.ca/server/oai/request"},"display":{"title":"Algorithms for Regular Chains of Dimension One","abstract":"One of the core commands in the RegularChains library inside Maple is Triangularize. The underlying decomposes the solution set of a polynomial system into geometrically meaningful components represented by regular chains. This algorithm works by repeatedly calling a procedure, called Intersect, which computes the common zeros of a polynomial p and a regular chain T . As the number of variables of p and T , as well as their degrees, increase, the call to the function Intersect(p, T ) becomes more and more computationally expensive. It was observed in that when the input polynomial system is zero-dimensional and T is one-dimensional then this cost can be substantially reduced. The method proposed by the authors is a probabilistic algorithm based on evaluation and interpolation techniques. This is the type of method which is typically challenging to implement in a high-level language like Maple’s language, as a sharp control of computing resources (in particular memory) is needed. In this document, we report on a successful implementation of this algorithm in Maple as well as in the BPAS library. On the other hand, multivariate Laurent series are a generalization of multivariate power series. They play an important roll when defining Puiseux series, and in consequence, they are required to implement Nowak’s version of the famous Newton-Puiseux algorithm. In this document, we also report a first implementation of a Laurent series object inside Maple together with the challenges that we have encountered during its development.","abstract_html":"One of the core commands in the RegularChains library inside Maple is Triangularize. The underlying decomposes the solution set of a polynomial system into geometrically meaningful components represented by regular chains. This algorithm works by repeatedly calling a procedure, called Intersect, which computes the common zeros of a polynomial p and a regular chain T . As the number of variables of p and T , as well as their degrees, increase, the call to the function Intersect(p, T ) becomes more and more computationally expensive. It was observed in that when the input polynomial system is zero-dimensional and T is one-dimensional then this cost can be substantially reduced. The method proposed by the authors is a probabilistic algorithm based on evaluation and interpolation techniques. This is the type of method which is typically challenging to implement in a high-level language like Maple’s language, as a sharp control of computing resources (in particular memory) is needed. In this document, we report on a successful implementation of this algorithm in Maple as well as in the BPAS library. On the other hand, multivariate Laurent series are a generalization of multivariate power series. They play an important roll when defining Puiseux series, and in consequence, they are required to implement Nowak’s version of the famous Newton-Puiseux algorithm. In this document, we also report a first implementation of a Laurent series object inside Maple together with the challenges that we have encountered during its development.","abstract_has_math":false,"creators":["Gonzalez Trochez, Juan P"],"institution":"The University of Western Ontario","degree_name":"M Sc","degree_level":null,"degree_discipline":"Computer Science","degree_department":null,"school":null,"contributors":[],"advisors":["Moreno Maza, Marc"],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-04-21","date_published":"2022-04-21","updated_at":"2026-07-27T21:55:56Z","subjects":["Regular chains","modular method","evaluation and interpolation","Laurent series","power series","subresultant chains"],"languages":["en_ca"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/20.500.14721/32042","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Moreno Maza, Marc"]},{"key":"dc:creator","label":"Author","values":["Gonzalez Trochez, Juan P"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-07-10T19:26:04Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2025-07-10T19:26:04Z"]},{"key":"dc:date.issued","label":"Date","values":["2022-04-21"]},{"key":"dc:publisher","label":"Institution","values":["The University of Western Ontario"]},{"key":"dc:type","label":"Dc Type","values":["thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Computer Science"]},{"key":"thesis:degree_name","label":"Degree Name","values":["M Sc"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Regular chains","modular method","evaluation and interpolation","Laurent series","power series","subresultant chains"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_ca"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/20.500.14721/32042"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The thesis cover page in the PDF document includes references to Western University’s previous institutional repository platform, known as Scholarship@Western, and links to that platform (beginning with ir.lib.uwo.ca). In citing or referring to this thesis, use the DOI or handle from this page instead. Sample citation: Author name, \"Thesis title.\" (Year). Western University Open Repository. https://doi.org/10.71858/123456."]},{"key":"dc:description.abstract","label":"Abstract","values":["One of the core commands in the RegularChains library inside Maple is Triangularize. The underlying decomposes the solution set of a polynomial system into geometrically meaningful components represented by regular chains. This algorithm works by repeatedly calling a procedure, called Intersect, which computes the common zeros of a polynomial p and a regular chain T . As the number of variables of p and T , as well as their degrees, increase, the call to the function Intersect(p, T ) becomes more and more computationally expensive. It was observed in that when the input polynomial system is zero-dimensional and T is one-dimensional then this cost can be substantially reduced. The method proposed by the authors is a probabilistic algorithm based on evaluation and interpolation techniques. This is the type of method which is typically challenging to implement in a high-level language like Maple’s language, as a sharp control of computing resources (in particular memory) is needed. In this document, we report on a successful implementation of this algorithm in Maple as well as in the BPAS library. On the other hand, multivariate Laurent series are a generalization of multivariate power series. They play an important roll when defining Puiseux series, and in consequence, they are required to implement Nowak’s version of the famous Newton-Puiseux algorithm. In this document, we also report a first implementation of a Laurent series object inside Maple together with the challenges that we have encountered during its development."]},{"key":"dc:title","label":"Title","values":["Algorithms for Regular Chains of Dimension One"]}]}],"canonical_facts":{"dc:contributor.advisor":["Moreno Maza, Marc"],"dc:creator":["Gonzalez Trochez, Juan P"],"dc:date.accessioned":["2025-07-10T19:26:04Z"],"dc:date.available":["2025-07-10T19:26:04Z"],"dc:date.issued":["2022-04-21"],"dc:description":["The thesis cover page in the PDF document includes references to Western University’s previous institutional repository platform, known as Scholarship@Western, and links to that platform (beginning with ir.lib.uwo.ca). In citing or referring to this thesis, use the DOI or handle from this page instead. Sample citation: Author name, \"Thesis title.\" (Year). Western University Open Repository. https://doi.org/10.71858/123456."],"dc:description.abstract":["One of the core commands in the RegularChains library inside Maple is Triangularize. The underlying decomposes the solution set of a polynomial system into geometrically meaningful components represented by regular chains. This algorithm works by repeatedly calling a procedure, called Intersect, which computes the common zeros of a polynomial p and a regular chain T . As the number of variables of p and T , as well as their degrees, increase, the call to the function Intersect(p, T ) becomes more and more computationally expensive. It was observed in that when the input polynomial system is zero-dimensional and T is one-dimensional then this cost can be substantially reduced. The method proposed by the authors is a probabilistic algorithm based on evaluation and interpolation techniques. This is the type of method which is typically challenging to implement in a high-level language like Maple’s language, as a sharp control of computing resources (in particular memory) is needed. In this document, we report on a successful implementation of this algorithm in Maple as well as in the BPAS library. On the other hand, multivariate Laurent series are a generalization of multivariate power series. They play an important roll when defining Puiseux series, and in consequence, they are required to implement Nowak’s version of the famous Newton-Puiseux algorithm. 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