{"id":4354,"date":"2026-09-17T14:57:11","date_gmt":"2026-09-17T21:57:11","guid":{"rendered":"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&#038;p=4354"},"modified":"2026-09-17T14:57:11","modified_gmt":"2026-09-17T21:57:11","slug":"hilbert-basis-for-conical-semigroups-and-indecomposable-elements-lenny-fukshansky-cmc","status":"publish","type":"tribe_events","link":"https:\/\/colleges.claremont.edu\/ccms\/event\/hilbert-basis-for-conical-semigroups-and-indecomposable-elements-lenny-fukshansky-cmc\/","title":{"rendered":"Hilbert basis for conical semigroups and indecomposable elements (Lenny Fukshansky, CMC)"},"content":{"rendered":"<p>The unique minimal generating set for a semigroup of lattice points in an n-dimensional polyhedral cone is called its Hilbert basis. In the case when the cone is spanned by lattice points, this Hilbert basis is finite, however its cardinality can be much larger than n. Nevertheless, a conjecture of Seb\u00f6 states that each point in this semigroup can be spanned by no more than n elements of the Hilbert basis. We will brief discuss this conjecture and its current status, and then shift to the situation where the cone in question is the positive orthant in R^n. In this case the Hilbert basis consists of the so-called indecomposable elements, notation coming from the theory of universal quadratic forms over algebraic number fields. We will classify lattices that have finite versus infinite Hilbert basis of indecomposables. We will then discuss an analogue of Seb\u00f6&#8217;s conjecture for indecomposables in planar lattices and finish by presenting a counting estimate on the number of indecomposables of bounded norm in this 2-dimensional case. Joint work with Filiana Kostopoulou.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The unique minimal generating set for a semigroup of lattice points in an n-dimensional polyhedral cone is called its Hilbert basis. In the case when the cone is spanned by [&hellip;]<\/p>\n","protected":false},"author":73,"featured_media":0,"template":"","meta":{"_acf_changed":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":""},"tags":[],"tribe_events_cat":[13],"class_list":["post-4354","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 v28.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Hilbert basis for conical semigroups and indecomposable elements (Lenny Fukshansky, CMC) - 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\/hilbert-basis-for-conical-semigroups-and-indecomposable-elements-lenny-fukshansky-cmc\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Hilbert basis for conical semigroups and indecomposable elements (Lenny Fukshansky, CMC) - Claremont Center for the Mathematical Sciences\" \/>\n<meta property=\"og:description\" content=\"The unique minimal generating set for a semigroup of lattice points in an n-dimensional polyhedral cone is called its Hilbert basis. In the case when the cone is spanned by [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/colleges.claremont.edu\/ccms\/event\/hilbert-basis-for-conical-semigroups-and-indecomposable-elements-lenny-fukshansky-cmc\/\" \/>\n<meta property=\"og:site_name\" content=\"Claremont Center for the Mathematical Sciences\" \/>\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\\\/hilbert-basis-for-conical-semigroups-and-indecomposable-elements-lenny-fukshansky-cmc\\\/\",\"url\":\"https:\\\/\\\/colleges.claremont.edu\\\/ccms\\\/event\\\/hilbert-basis-for-conical-semigroups-and-indecomposable-elements-lenny-fukshansky-cmc\\\/\",\"name\":\"Hilbert basis for conical semigroups and indecomposable elements (Lenny Fukshansky, CMC) - Claremont Center for the Mathematical Sciences\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/colleges.claremont.edu\\\/ccms\\\/#website\"},\"datePublished\":\"2026-09-17T21:57:11+00:00\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/colleges.claremont.edu\\\/ccms\\\/event\\\/hilbert-basis-for-conical-semigroups-and-indecomposable-elements-lenny-fukshansky-cmc\\\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/colleges.claremont.edu\\\/ccms\\\/event\\\/hilbert-basis-for-conical-semigroups-and-indecomposable-elements-lenny-fukshansky-cmc\\\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/colleges.claremont.edu\\\/ccms\\\/event\\\/hilbert-basis-for-conical-semigroups-and-indecomposable-elements-lenny-fukshansky-cmc\\\/#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\":\"Hilbert basis for conical semigroups and indecomposable elements (Lenny Fukshansky, CMC)\"}]},{\"@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":"Hilbert basis for conical semigroups and indecomposable elements (Lenny Fukshansky, CMC) - 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\/hilbert-basis-for-conical-semigroups-and-indecomposable-elements-lenny-fukshansky-cmc\/","og_locale":"en_US","og_type":"article","og_title":"Hilbert basis for conical semigroups and indecomposable elements (Lenny Fukshansky, CMC) - Claremont Center for the Mathematical Sciences","og_description":"The unique minimal generating set for a semigroup of lattice points in an n-dimensional polyhedral cone is called its Hilbert basis. 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