{"id":1599,"date":"2019-10-08T13:37:42","date_gmt":"2019-10-08T20:37:42","guid":{"rendered":"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&#038;p=1599"},"modified":"2020-01-20T11:54:43","modified_gmt":"2020-01-20T19:54:43","slug":"antc-seminar-sinai-robins-university-of-sao-paulo","status":"publish","type":"tribe_events","link":"https:\/\/colleges.claremont.edu\/ccms\/event\/antc-seminar-sinai-robins-university-of-sao-paulo\/","title":{"rendered":"Dragging the roots of a polynomial to the unit circle (Sinai Robins, University of Sao Paulo)"},"content":{"rendered":"<div>Several conditions are known for a self-inversive polynomial that ascertain the location of its roots, and we present a framework for comparison of those conditions. We associate a parametric family of polynomials p_\u03b1(x) to each such polynomial p, and define cn(p), il(p) to be the sharp threshold values of \u03b1 that guarantee that, for all larger values of the parameter, p_\u03b1(x) has, respectively, all roots in the unit circle and all roots interlacing the roots of unity of the same degree.\u00a0 Interlacing implies circle rootedness, hence il(p) \u2265 cn(p), and this inequality is often used for showing circle rootedness. Both il(p) and cn(p) turn out to be semi-algebraic functions of the coefficients of p, and some useful bounds are also presented, entailing several known results about roots in the circle. The study of il(p) leads to a rich classification of real self-inversive polynomials of each degree, organizing them into a complete polyhedral fan. We have a close look at the class of polynomials for which il(p) = cn(p), whereas in general the quotient il(p)\/cn(p) is shown to be unbounded as the degree grows. Several examples and open questions are presented.\u00a0 This is joint work with Arnaldo Mandel.<\/div>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n","protected":false},"excerpt":{"rendered":"<p>Several conditions are known for a self-inversive polynomial that ascertain the location of its roots, and we present a framework for comparison of those conditions. We associate a parametric family [&hellip;]<\/p>\n","protected":false},"author":73,"featured_media":0,"template":"","meta":{"_acf_changed":false,"_price":"","_stock":"","_tribe_ticket_header":"","_tribe_default_ticket_provider":"","_tribe_ticket_capacity":"0","_ticket_start_date":"","_ticket_end_date":"","_tribe_ticket_show_description":"","_tribe_ticket_show_not_going":false,"_tribe_ticket_use_global_stock":"","_tribe_ticket_global_stock_level":"","_global_stock_mode":"","_global_stock_cap":"","_tribe_rsvp_for_event":"","_tribe_ticket_going_count":"","_tribe_ticket_not_going_count":"","_tribe_tickets_list":"[]","_tribe_ticket_has_attendee_info_fields":false,"_tribe_events_status":"","_tribe_events_status_reason":"","_tribe_events_is_hybrid":"","_tribe_events_is_virtual":"","_tribe_events_virtual_video_source":"","_tribe_events_virtual_embed_video":"","_tribe_events_virtual_linked_button_text":"","_tribe_events_virtual_linked_button":"","_tribe_events_virtual_show_embed_at":"","_tribe_events_virtual_show_embed_to":[],"_tribe_events_virtual_show_on_event":"","_tribe_events_virtual_show_on_views":"","_tribe_events_virtual_url":"","footnotes":"","_tec_slr_enabled":"","_tec_slr_layout":""},"tags":[],"tribe_events_cat":[13],"class_list":["post-1599","tribe_events","type-tribe_events","status-publish","hentry","tribe_events_cat-antc-seminar","cat_antc-seminar"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.2 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Dragging the roots of a polynomial to the unit circle (Sinai Robins, University of Sao Paulo) - Claremont Center for the Mathematical Sciences<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/colleges.claremont.edu\/ccms\/event\/antc-seminar-sinai-robins-university-of-sao-paulo\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Dragging the roots of a polynomial to the unit circle (Sinai Robins, University of Sao Paulo) - Claremont Center for the Mathematical Sciences\" \/>\n<meta property=\"og:description\" content=\"Several conditions are known for a self-inversive polynomial that ascertain the location of its roots, and we present a framework for comparison of those conditions. We associate a parametric family [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/colleges.claremont.edu\/ccms\/event\/antc-seminar-sinai-robins-university-of-sao-paulo\/\" \/>\n<meta property=\"og:site_name\" content=\"Claremont Center for the Mathematical Sciences\" \/>\n<meta property=\"article:modified_time\" content=\"2020-01-20T19:54:43+00:00\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data1\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/colleges.claremont.edu\/ccms\/event\/antc-seminar-sinai-robins-university-of-sao-paulo\/\",\"url\":\"https:\/\/colleges.claremont.edu\/ccms\/event\/antc-seminar-sinai-robins-university-of-sao-paulo\/\",\"name\":\"Dragging the roots of a polynomial to the unit circle (Sinai Robins, University of Sao Paulo) - Claremont Center for the Mathematical Sciences\",\"isPartOf\":{\"@id\":\"https:\/\/colleges.claremont.edu\/ccms\/#website\"},\"datePublished\":\"2019-10-08T20:37:42+00:00\",\"dateModified\":\"2020-01-20T19:54:43+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/colleges.claremont.edu\/ccms\/event\/antc-seminar-sinai-robins-university-of-sao-paulo\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/colleges.claremont.edu\/ccms\/event\/antc-seminar-sinai-robins-university-of-sao-paulo\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/colleges.claremont.edu\/ccms\/event\/antc-seminar-sinai-robins-university-of-sao-paulo\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/colleges.claremont.edu\/ccms\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Events\",\"item\":\"https:\/\/colleges.claremont.edu\/ccms\/events\/\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"Dragging the roots of a polynomial to the unit circle (Sinai Robins, University of Sao Paulo)\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/colleges.claremont.edu\/ccms\/#website\",\"url\":\"https:\/\/colleges.claremont.edu\/ccms\/\",\"name\":\"Claremont Center for the Mathematical Sciences\",\"description\":\"Proudly Serving the Math Community at the Claremont Colleges Since 2007\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/colleges.claremont.edu\/ccms\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Dragging the roots of a polynomial to the unit circle (Sinai Robins, University of Sao Paulo) - Claremont Center for the Mathematical Sciences","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/colleges.claremont.edu\/ccms\/event\/antc-seminar-sinai-robins-university-of-sao-paulo\/","og_locale":"en_US","og_type":"article","og_title":"Dragging the roots of a polynomial to the unit circle (Sinai Robins, University of Sao Paulo) - Claremont Center for the Mathematical Sciences","og_description":"Several conditions are known for a self-inversive polynomial that ascertain the location of its roots, and we present a framework for comparison of those conditions. 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