{"id":396380,"date":"2019-01-17T09:55:57","date_gmt":"2019-01-17T17:55:57","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&#038;p=396380"},"modified":"2019-01-17T09:55:57","modified_gmt":"2019-01-17T17:55:57","slug":"efficient-method-computing-resultant-systems","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/efficient-method-computing-resultant-systems\/","title":{"rendered":"An Efficient Method For Computing Resultant Systems"},"content":{"rendered":"<section id=\"Abs1\" class=\"Abstract\" lang=\"en\" tabindex=\"-1\">\n<p class=\"Para\">A resultant system of a finite set of polynomials in two homogeneous variables is a finite set of polynomials in the coefficients of the original polynomials whose vanishing is a necessary and sufficient condition for the original polynomials to have a common non-trivial zero. We present an efficient general method for computing resultant systems. If there are\u00a0<em class=\"EmphasisTypeItalic \">s<\/em>\u00a0polynomials and each has degree at most\u00a0<em class=\"EmphasisTypeItalic \">d<\/em>, the resultant system will have\u00a0<em class=\"EmphasisTypeItalic \">O<\/em>(<em class=\"EmphasisTypeItalic \">s<\/em><sup><em class=\"EmphasisTypeItalic \">2<\/em><\/sup><em class=\"EmphasisTypeItalic \">d<\/em>) polynomials, each of which will have degree at most\u00a0<em class=\"EmphasisTypeItalic \">d<\/em><sup><em class=\"EmphasisTypeItalic \">2<\/em><\/sup>.<\/p>\n<\/section>\n","protected":false},"excerpt":{"rendered":"<p>A resultant system of a finite set of polynomials in two homogeneous variables is a finite set of polynomials in the coefficients of the original polynomials whose vanishing is a necessary and sufficient condition for the original polynomials to have a common non-trivial zero. We present an efficient general method for computing resultant systems. If [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13561,13546],"msr-publication-type":[193715],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-396380","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"Springer-Verlag","msr_edition":"","msr_affiliation":"","msr_published_date":"1998-11-01","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"243\u2013245","msr_chapter":"","msr_isbn":"","msr_journal":"Applicable Algebra in Engineering, Communication and Computing (AAECC), Issue 3","msr_volume":"9","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"","msr_publicationurl":"https:\/\/link.springer.com\/article\/10.1007\/s002000050105","msr_doi":"10.1007\/s002000050105","msr_publication_uploader":[{"type":"url","title":"https:\/\/link.springer.com\/article\/10.1007\/s002000050105","viewUrl":false,"id":false,"label_id":0},{"type":"doi","title":"10.1007\/s002000050105","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_attachments":[{"id":0,"url":"https:\/\/link.springer.com\/article\/10.1007\/s002000050105"}],"msr-author-ordering":[{"type":"user_nicename","value":"markenc","user_id":32814,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=markenc"}],"msr_impact_theme":[],"msr_research_lab":[],"msr_event":[],"msr_group":[493619],"msr_project":[],"publication":[],"video":[],"download":[],"msr_publication_type":"article","related_content":[],"_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/396380"}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-research-item"}],"version-history":[{"count":4,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/396380\/revisions"}],"predecessor-version":[{"id":562437,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/396380\/revisions\/562437"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=396380"}],"wp:term":[{"taxonomy":"msr-content-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-content-type?post=396380"},{"taxonomy":"msr-research-highlight","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-highlight?post=396380"},{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=396380"},{"taxonomy":"msr-publication-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-publication-type?post=396380"},{"taxonomy":"msr-product-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-product-type?post=396380"},{"taxonomy":"msr-focus-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-focus-area?post=396380"},{"taxonomy":"msr-platform","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-platform?post=396380"},{"taxonomy":"msr-download-source","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-download-source?post=396380"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=396380"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=396380"},{"taxonomy":"msr-field-of-study","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-field-of-study?post=396380"},{"taxonomy":"msr-conference","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-conference?post=396380"},{"taxonomy":"msr-journal","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-journal?post=396380"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=396380"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=396380"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}