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"//netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css", "//fonts.googleapis.com/css?family=Open+Sans:400,300,600,700|Merriweather:400,300,700", "#", "//netdna.bootstrapcdn.com/bootstrap/3.1.1/js/bootstrap.min.js", "#maincontent", "/en/person/matsa", "/css/cookieconsent.min.css", "/scripts/cookieconsent.min.js", "https://www.lib.chalmers.se/en/", "//cdnjs.cloudflare.com/ajax/libs/select2/4.0.1/js/select2.min.js", "/en/organization/?tab=projects\u0026f_funder=0d84752e-ee44-485f-b889-bcbe3cf6b095|Swedish Research Council (VR)", "/en/accessibility/", "https://odr.chalmers.se", "https://citationstyles.org/", "/Account/Login?returnUrl=/en/project/6747", "https://research.chalmers.se/en/project/6747", "//cdnjs.cloudflare.com/ajax/libs/jquery.tablesorter/2.16.4/jquery.tablesorter.min.js", "/en/", "/css/bootstrap-datetimepicker.min.css", "//code.jquery.com/jquery-1.11.0.min.js" ], "httpBodyMetaTags": { "description": "Teorin för geometri går tillbaks till antiken, men först på 1600-talet infördes systematisk representation av geometriska objekt, så kallade varieteter, såsom linjer och kurvor etcetera, som lösningar till olika ekvationer. Till exempel kan en cirkel i planet representeras som mängden av punkter i planet där ett visst polynom p(x,y) av två reella variabler är noll. Detta visade sig mycket fruktbart och öppnade vägen för den matematiska analysen som grundlades av Newton and Leibniz och som är oumbärlig för nästan all modern naturvetenskap. Det visar sig praktiskt att gå ett steg längre och representera cirkeln med mängden av alla olika polynom p(x,y) som är noll på cirkeln. Man får på detta sätt ett s.k. ideal av polynom som representerar cirkeln. Å andra sidan finns det ideal som inte uppkommer på detta sätt; de kan då tolkas som ett slags generaliserade geometriska objekt. Detta är grundiden i s.k. algebraisk geometri. För att vara mer precis så studerar man ofta polynom av komplexa variabler men iden är densamma. Ett syfte med denna verksamhet är att skaffa sig fördjupad förståelse av lösningar till system av polynomekvationer. Det finns ett viktigt begrepp den matematiska analysen, s.k. strömmar, som kan ses som en slags generaliserade funktioner. Det finns en ström som är noll överallt utom på cirkeln ifråga, och sådan att om man multiplicerar den med ett polynom så får man noll om och endast om polynomet själv är noll på cirkeln. På detta sätt kan man säga att även strömmen representerar cirkeln. I ett tidigare projekt har vi visat att det till varje ideal finns en uppsättning strömmar sådan att ett polynom ligger i idealet om och endast om produkten av polynomet och var och en av strömmarna blir noll. Ett ideal är ofta givet genom att man har en uppsättning polynom f_1,..., f_m, där idealet då är alla polynom P som går att uttrycka som en summa P=f_1 u_1+... +f_m u_m, där u_1,..., u_m är andra polynom. Polynomen f_1,..., f_m sägs då vara generatorer för idealet. Det är en grundläggande frågeställning att givet en uppsättning generatorer avgöra om ett givet polynom P ligger i idealet, dvs om man kan lösa ekvationen ovan. Detta kallas ``membership problem´´ och man kan säga att vi in viss mening med vår representation med strömmar har löst detta problem för ett tämligen allmänt ideal. Vi kan tom skriva upp en formel som ger en uppsättning polynom u_j som löser ekvationen givet att P ligger i idealet. Det nya projektet består av tre delar. En del handlar om att studera ett liknande idealproblem som ovan, fast man vill bara lösa ekvationen så att den gäller bara för sådana punkter som ligger på en delvarietet (detta är ett geometriskt objekt i sig, t ex en cirkel). Detta erbjuder inte några egentliga nya svårigheter om den delvarietet man befinner sig på är glatt, som tex cirkeln; att den är glatt betyder att om man tittar väldigt lokalt, med förstoringsglas nära en punkt, så ser den i allt väsentligt ser ut som en bit av en linje. Om man å andra sidan tittar på tex den varietet som definieras av ekvationen x^3-y^2=0 så ser den inte alls ut som en linje i närheten av punkten (0,0), hur mycket man än förstorar; man säger att denna varietet har en singularitet vid punkten (0,0). Just denna varietet brukar kallas en spets. I annan del handlar det istället om att lösa en viss partiell differentialekvation, den s.k. dbar-ekvationen som är av central betydelse för komplex analys och algebraisk geometri, på en singular varietet. Igen är det så att denna ekvation är väl studerad på varieteter utan singulariteter, men inte särskilt väl analyserad när varieten har singularieter. Den tredje delen av projektet handlar om att studera snitt (eller skärning) av varieteter. Om man tar snittet av en cirkel och en linje så får man i allmänhet två olika punkter (eller inga alls, och om linjen precis tangerar cirkeln får man en punkt). Det blir knepigare om man tar snittet av en linje och spetsen. Det finns en väl utvecklad teori om hur snittet av singulära varieteter ska tolkas men i allmänhet kan man inte säga att snittet är en bestämd varietet utan bara en s.k. ekvivalensklass av varieteter (eller noga taget cykler, som är ett lite generellare begrepp). Projektet går ut på att i många intressanta fall trots allt hitta en specifik representation av snittet av varieteter i form av en ström.", "og:description": "Teorin för geometri går tillbaks till antiken, men först på 1600-talet infördes systematisk representation av geometriska objekt, så kallade varieteter, såsom linjer och kurvor etcetera, som lösningar till olika ekvationer. Till exempel kan en cirkel i planet representeras som mängden av punkter i planet där ett visst polynom p(x,y) av två reella variabler är noll. Detta visade sig mycket fruktbart och öppnade vägen för den matematiska analysen som grundlades av Newton and Leibniz och som är oumbärlig för nästan all modern naturvetenskap. Det visar sig praktiskt att gå ett steg längre och representera cirkeln med mängden av alla olika polynom p(x,y) som är noll på cirkeln. Man får på detta sätt ett s.k. ideal av polynom som representerar cirkeln. Å andra sidan finns det ideal som inte uppkommer på detta sätt; de kan då tolkas som ett slags generaliserade geometriska objekt. Detta är grundiden i s.k. algebraisk geometri. För att vara mer precis så studerar man ofta polynom av komplexa variabler men iden är densamma. Ett syfte med denna verksamhet är att skaffa sig fördjupad förståelse av lösningar till system av polynomekvationer. Det finns ett viktigt begrepp den matematiska analysen, s.k. strömmar, som kan ses som en slags generaliserade funktioner. Det finns en ström som är noll överallt utom på cirkeln ifråga, och sådan att om man multiplicerar den med ett polynom så får man noll om och endast om polynomet själv är noll på cirkeln. På detta sätt kan man säga att även strömmen representerar cirkeln. I ett tidigare projekt har vi visat att det till varje ideal finns en uppsättning strömmar sådan att ett polynom ligger i idealet om och endast om produkten av polynomet och var och en av strömmarna blir noll. Ett ideal är ofta givet genom att man har en uppsättning polynom f_1,..., f_m, där idealet då är alla polynom P som går att uttrycka som en summa P=f_1 u_1+... +f_m u_m, där u_1,..., u_m är andra polynom. Polynomen f_1,..., f_m sägs då vara generatorer för idealet. Det är en grundläggande frågeställning att givet en uppsättning generatorer avgöra om ett givet polynom P ligger i idealet, dvs om man kan lösa ekvationen ovan. Detta kallas ``membership problem´´ och man kan säga att vi in viss mening med vår representation med strömmar har löst detta problem för ett tämligen allmänt ideal. Vi kan tom skriva upp en formel som ger en uppsättning polynom u_j som löser ekvationen givet att P ligger i idealet. Det nya projektet består av tre delar. En del handlar om att studera ett liknande idealproblem som ovan, fast man vill bara lösa ekvationen så att den gäller bara för sådana punkter som ligger på en delvarietet (detta är ett geometriskt objekt i sig, t ex en cirkel). Detta erbjuder inte några egentliga nya svårigheter om den delvarietet man befinner sig på är glatt, som tex cirkeln; att den är glatt betyder att om man tittar väldigt lokalt, med förstoringsglas nära en punkt, så ser den i allt väsentligt ser ut som en bit av en linje. Om man å andra sidan tittar på tex den varietet som definieras av ekvationen x^3-y^2=0 så ser den inte alls ut som en linje i närheten av punkten (0,0), hur mycket man än förstorar; man säger att denna varietet har en singularitet vid punkten (0,0). Just denna varietet brukar kallas en spets. I annan del handlar det istället om att lösa en viss partiell differentialekvation, den s.k. dbar-ekvationen som är av central betydelse för komplex analys och algebraisk geometri, på en singular varietet. Igen är det så att denna ekvation är väl studerad på varieteter utan singulariteter, men inte särskilt väl analyserad när varieten har singularieter. Den tredje delen av projektet handlar om att studera snitt (eller skärning) av varieteter. Om man tar snittet av en cirkel och en linje så får man i allmänhet två olika punkter (eller inga alls, och om linjen precis tangerar cirkeln får man en punkt). Det blir knepigare om man tar snittet av en linje och spetsen. Det finns en väl utvecklad teori om hur snittet av singulära varieteter ska tolkas men i allmänhet kan man inte säga att snittet är en bestämd varietet utan bara en s.k. ekvivalensklass av varieteter (eller noga taget cykler, som är ett lite generellare begrepp). Projektet går ut på att i många intressanta fall trots allt hitta en specifik representation av snittet av varieteter i form av en ström.", "twitter:description": "Teorin för geometri går tillbaks till antiken, men först på 1600-talet infördes systematisk representation av geometriska objekt, så kallade varieteter, såsom linjer och kurvor etcetera, som lösningar till olika ekvationer. Till exempel kan en cirkel i planet representeras som mängden av punkter i planet där ett visst polynom p(x,y) av två reella variabler är noll. Detta visade sig mycket fruktbart och öppnade vägen för den matematiska analysen som grundlades av Newton and Leibniz och som är oumbärlig för nästan all modern naturvetenskap. Det visar sig praktiskt att gå ett steg längre och representera cirkeln med mängden av alla olika polynom p(x,y) som är noll på cirkeln. Man får på detta sätt ett s.k. ideal av polynom som representerar cirkeln. Å andra sidan finns det ideal som inte uppkommer på detta sätt; de kan då tolkas som ett slags generaliserade geometriska objekt. Detta är grundiden i s.k. algebraisk geometri. För att vara mer precis så studerar man ofta polynom av komplexa variabler men iden är densamma. Ett syfte med denna verksamhet är att skaffa sig fördjupad förståelse av lösningar till system av polynomekvationer. Det finns ett viktigt begrepp den matematiska analysen, s.k. strömmar, som kan ses som en slags generaliserade funktioner. Det finns en ström som är noll överallt utom på cirkeln ifråga, och sådan att om man multiplicerar den med ett polynom så får man noll om och endast om polynomet själv är noll på cirkeln. På detta sätt kan man säga att även strömmen representerar cirkeln. I ett tidigare projekt har vi visat att det till varje ideal finns en uppsättning strömmar sådan att ett polynom ligger i idealet om och endast om produkten av polynomet och var och en av strömmarna blir noll. Ett ideal är ofta givet genom att man har en uppsättning polynom f_1,..., f_m, där idealet då är alla polynom P som går att uttrycka som en summa P=f_1 u_1+... +f_m u_m, där u_1,..., u_m är andra polynom. Polynomen f_1,..., f_m sägs då vara generatorer för idealet. Det är en grundläggande frågeställning att givet en uppsättning generatorer avgöra om ett givet polynom P ligger i idealet, dvs om man kan lösa ekvationen ovan. Detta kallas ``membership problem´´ och man kan säga att vi in viss mening med vår representation med strömmar har löst detta problem för ett tämligen allmänt ideal. Vi kan tom skriva upp en formel som ger en uppsättning polynom u_j som löser ekvationen givet att P ligger i idealet. Det nya projektet består av tre delar. En del handlar om att studera ett liknande idealproblem som ovan, fast man vill bara lösa ekvationen så att den gäller bara för sådana punkter som ligger på en delvarietet (detta är ett geometriskt objekt i sig, t ex en cirkel). Detta erbjuder inte några egentliga nya svårigheter om den delvarietet man befinner sig på är glatt, som tex cirkeln; att den är glatt betyder att om man tittar väldigt lokalt, med förstoringsglas nära en punkt, så ser den i allt väsentligt ser ut som en bit av en linje. Om man å andra sidan tittar på tex den varietet som definieras av ekvationen x^3-y^2=0 så ser den inte alls ut som en linje i närheten av punkten (0,0), hur mycket man än förstorar; man säger att denna varietet har en singularitet vid punkten (0,0). Just denna varietet brukar kallas en spets. I annan del handlar det istället om att lösa en viss partiell differentialekvation, den s.k. dbar-ekvationen som är av central betydelse för komplex analys och algebraisk geometri, på en singular varietet. Igen är det så att denna ekvation är väl studerad på varieteter utan singulariteter, men inte särskilt väl analyserad när varieten har singularieter. Den tredje delen av projektet handlar om att studera snitt (eller skärning) av varieteter. Om man tar snittet av en cirkel och en linje så får man i allmänhet två olika punkter (eller inga alls, och om linjen precis tangerar cirkeln får man en punkt). Det blir knepigare om man tar snittet av en linje och spetsen. Det finns en väl utvecklad teori om hur snittet av singulära varieteter ska tolkas men i allmänhet kan man inte säga att snittet är en bestämd varietet utan bara en s.k. ekvivalensklass av varieteter (eller noga taget cykler, som är ett lite generellare begrepp). Projektet går ut på att i många intressanta fall trots allt hitta en specifik representation av snittet av varieteter i form av en ström." }, "httpBodyByteSize": 46525, "httpBodyPageTitle": "Residue theory on singular spaces and algebraic geometry" }, "/en/publication/25191": { "date": "2026-02-06T19:42:56Z", "httpProtocol": "h2, tls/1.3", "httpStatus": 200, "ipAddress": "40.113.65.9", "ipASN": "Microsoft Corporation", "ipCountry": "Ireland", "httpHeaderHash": "MRT2HF5XML76NNMFH77LU3RKS2GIYTGP", "httpHeaderTechnologies": [ "HSTS", "Azure", "IIS:10.0", "Windows Server", "Microsoft ASP.NET" ], "httpHeaderUrls": [], "httpHeaderByteSize": 905, "httpBodyHash": "E7QNQEAZTGE44G4P65XWP2FNSHOGMX5K", "httpBodyTechnologies": [ "jQuery CDN", "BootstrapCDN:3.1.1", "Tablesorter:2.16.4", "Onsen UI", "Cloudflare", "jQuery:1.11.0", "Bootstrap:3.1.1", "cdnjs", "Lodash", "Select2", "TinyMCE" ], "httpBodyUrls": [ "https://github.com/Juris-M/citeproc-js", 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