Harnack's curve theorem

"M-curve" redirects here. For a curve describing the demographics of working women, see Career woman § Limited work options.
The elliptic curve (smooth degree 3) on the left is an M-curve, as it has the maximum (2) components, while the curve on the right has only 1 component.

In real algebraic geometry, Harnack's curve theorem, named after Axel Harnack, gives the possible numbers of connected components that an algebraic curve can have, in terms of the degree of the curve. For any algebraic curve of degree m in the real projective plane, the number of components c is bounded by

The maximum number is one more than the maximum genus of a curve of degree m, attained when the curve is nonsingular. Moreover, any number of components in this range of possible values can be attained.

The Trott curve, shown here with 7 of its bitangents, is a quartic (degree 4) M-curve, attaining the maximum (4) components for a curve of that degree.

A curve which attains the maximum number of real components is called an M-curve (from "maximum") – for example, an elliptic curve with two components, such as or the Trott curve, a quartic with four components, are examples of M-curves.

This theorem formed the background to Hilbert's sixteenth problem.

In a recent development a Harnack curve is shown to be a curve whose amoeba has area equal to the Newton polygon of the polynomial P, which is called the characteristic curve of dimer models, and every Harnack curve is the spectral curve of some dimer model.(Mikhalkin 2001)(Kenyon, Okounkov & Sheffield (2006))

References

  • Dmitrii Andreevich Gudkov, The topology of real projective algebraic varieties, Uspekhi Mat. Nauk 29 (1974), 3–79 (Russian), English transl., Russian Math. Surveys 29:4 (1974), 1–79
  • Carl Gustav Axel Harnack, Ueber die Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann. 10 (1876), 189–199
  • George Wilson, Hilbert's sixteenth problem, Topology 17 (1978), 53–74
  • Kenyon, Richard; Okounkov, Andrei; Sheffield, Scott (2006). "Dimers and Amoebae". Annals of Mathematics. 163 (3): 1019–1056. arXiv:math-ph/0311005. doi:10.4007/annals.2006.163.1019. MR 2215138. S2CID 119724053.
  • Mikhalkin, Grigory (2001), Amoebas of algebraic varieties, arXiv:math/0108225, MR 2102998

Read other articles:

Наследники: Недобрый мир

Значимость предмета статьи поставлена под сомнение.Пожалуйста, покажите в статье значимость её предмета, добавив в неё доказательства значимости по частным критериям значимости или, в случае если частные критерии значимости для предмета статьи отсутствуют, по об�...

 

Daftar katedral di Iran

Katedral Consolata, Teheran Katedral Santo Yosef, Teheran Ini adalah daftar katedral di Iran diurutkan berdasarkan denominasi. Katolik Katedral Gereja Katolik di Iran:[1] Katedral Consolata, Teheran (Gereja Latin) Gereja Surp Mesrob, Ahvaz (Gereja Katolik Kaldea) Gereja Santo Yaghou, Salmas (Gereja Katolik Kaldea) Katedral Santo Yosef, Teheran (Gereja Katolik Kaldea) Katedral Maria Bunda Allah, Urmia (Gereja Katolik Kaldea) Katedral Surp Grigor Lusavorich, Ispahan (Gereja Katolik Arme...

 

Холокост на Украине

Часть серии статей о Холокосте Идеология и политика Расовая гигиена · Расовый антисемитизм · Нацистская расовая политика · Нюрнбергские расовые законы Шоа Лагеря смерти Белжец · Дахау · Майданек · Малый Тростенец · Маутхаузен ·&...

Page Belcher

American politician (1899–1980) Page BelcherMember of the U.S. House of Representativesfrom OklahomaIn officeJanuary 3, 1951 – January 3, 1973Preceded byGeorge H. WilsonSucceeded byJames R. JonesConstituency8th district (1951–1953)1st district (1953–1973) Personal detailsBorn(1899-04-21)April 21, 1899Jefferson, Oklahoma TerritoryDiedAugust 2, 1980(1980-08-02) (aged 81)Midwest City, Oklahoma, U.S.Political partyRepublicanSpouseGladys CollinsAlma materUniversity of Okla...

 

العلاقات الأردنية الغامبية

العلاقات الأردنية الغامبية الأردن غامبيا   الأردن   غامبيا تعديل مصدري - تعديل   العلاقات الأردنية الغامبية هي العلاقات الثنائية التي تجمع بين الأردن وغامبيا.[1][2][3][4][5] مقارنة بين البلدين هذه مقارنة عامة ومرجعية للدولتين: وجه المقارنة الأ�...

 

X (TV series)

Japanese anime television series X TV redirects here. For the student television station, see XTV. XKey visual of the seriesGenreAction[1]Dark fantasy[1]Supernatural thriller[1][2] Anime television seriesDirected byYoshiaki KawajiriProduced byMasao MaruyamaJōichi MizunoMichiko SuzukiShinji KomoriWritten byYuki EnatsuMusic byNaoki SatōStudioMadhouseLicensed byAUS: Siren VisualNA: CrunchyrollUK: MVM FilmsOriginal networkWOWOWOriginal ru...

Seni Buddha-Yunani

Buddha dari Gandhara, abad pertama atau kedua Masehi. Seni Buddha-Yunani adalah bentuk manifestasi seni aliran Buddha-Yunani, sebuah perpaduan budaya antara budaya Yunani klasik dan agama Buddha, yang berkembang selama hampir 1.000 tahun di Asia Tengah, antara penaklukan oleh sang Alexander yang Agung pada abad ke-4 SM, dan penaklukan oleh orang-orang Islam pada abad ke-7. Seni Buddha-Yunani memiliki ciri khas realisme idealistik seni Yunani Helenis dan perwujudan pertama sang Buddha dalam be...

 

Rapidity

Measure of relativistic velocity Rapidity is the value of artanh(v / c) for velocity v and speed of light c Rapidity is a measure for relativistic velocity. For one-dimensional motion, rapidities are additive. However, velocities must be combined by Einstein's velocity-addition formula. For low speeds, rapidity and velocity are almost exactly proportional but, for higher velocities, rapidity takes a larger value, with the rapidity of light being infinite. Mathematically, rapidity can be defin...

 

Reparti mobili della Polizia di Stato

Voce principale: Polizia di Stato. Reparto mobileScudetto ricamato del repartoDescrizione generaleAttivo1946-oggi Nazione Italia Servizio Polizia di Stato TipoBattaglione CompitiPubblica sicurezzaOrdine pubblicoSoccorso Pubblico controllo antisommossa Mezzi in dotazioneAuto:Land Rover DiscoveryFiat Grande Punto, Fiat PandaFurgoni:Iveco DailyFiat DucatoBlindati:Iveco Eurocargo SoprannomeCeleriniCelere Anniversari27 maggio Sito internetSito ufficiale Parte del Ministero dell'interno Dipar...

Formula Satu musim 1990

Kejuaraan DuniaFormula Satu FIA 1990 Juara Dunia Pembalap: Ayrton Senna Juara Dunia Konstruktor: McLaren-Honda Sebelum: 1989 Sesudah: 1991 Balapan menurut negaraBalapan menurut musim Ayrton Senna berhasil memenangkan gelar juara dunia untuk yang kedua kalinya bersama dengan tim McLaren. Juara dunia bertahan, yaitu Alain Prost (foto tahun 2008), adalah runner-up, dengan membalap untuk tim Ferrari. Nelson Piquet (foto tahun 1983), mantan juara dunia sebanyak tiga kali, menyelesaikan musim ini ...

 

Dons (singer)

Latvian singer DonsDons performing at Summer Sound on 28 July 2017 at Jūrmalas Park in Liepāja, LatviaBackground informationBirth nameArtūrs ŠingirejsAlso known asArt Green, Art SingerBorn (1984-04-10) 10 April 1984 (age 40)Brocēni, LatviaGenresPop, pop rockOccupation(s) Singer songwriter Instrument(s)Vocals, guitar, pianoYears active2003–presentLabelsPlatforma Records, DonsMusic, Universal MusicMusical artist Artūrs Šingirejs (born 10 April 1984), better known as Dons, is a Lat...

 

سباق طواف فرنسا 1931

سباق طواف فرنسا 1931 الاسم سباق طواف فرنسا 1931 التاريخ 30 يونيو - 26 يوليو 1931 التاريخ بداية:30 يونيو 1931  نهاية:26 يوليو 1931  عدد المراحل 24 عدد الرياضيين 81 (نقطة البداية)،  و35 (نقطة النهاية)  المسافة 5091 الزمن 177 ساعة و10 دقيقة و03 ثانية البلد فرنسا موناكو سويسرا  نقطة البداية �...

Tugu Muda

Tugu MudaLandskap daerah Tugu MudaInformasi umumLokasiSemarang, JatengAlamatJalan Pemuda Jalan Pandanaran Jalan MGR SoegijopranotoMulai dibangun10 November 1951Rampung20 Mei 1953Tinggi53 meterDesain dan konstruksiArsitekDr Mundo Hendro Tugu Muda (bahasa Jawa: ꦠꦸꦒꦸ​ꦩꦸꦢ) adalah sebuah monumen yang dibuat untuk mengenang jasa-jasa para pahlawan yang telah gugur dalam Pertempuran Lima Hari di Semarang. Monumen ini terletak di Jalan Nasional Rute 20 yang mengarah ke Solo. Tugu ...

 

Hockey su prato

Hockey su pratoUna partita di hockey su pratoFederazioneFIH InventatoTra il 1863 ed il 1875 Componenti di una squadra11 GenereMaschile e femminile Indoor/outdoorOutdoor Campo di giocoCampo rettangolare OlimpicoSì Campione mondiale Germania (2023) Manuale L'hockey su prato è uno sport di squadra, variante della disciplina tradizionale.[1] Si disputa su campi in erba sintetica.[1] Le due squadre che si affrontano sono composte di 11 atleti, uno dei quali è il portiere. ...

 

哈萨克斯坦总统

此条目序言章节没有充分总结全文内容要点。 (2019年3月21日)请考虑扩充序言，清晰概述条目所有重點。请在条目的讨论页讨论此问题。 哈萨克斯坦總統哈薩克總統旗現任Қасым-Жомарт Кемелұлы Тоқаев卡瑟姆若马尔特·托卡耶夫自2019年3月20日在任任期7年首任努尔苏丹·纳扎尔巴耶夫设立1990年4月24日（哈薩克蘇維埃社會主義共和國總統） 哈萨克斯坦 哈萨克斯坦政府...

Daftar duta besar Uni Eropa untuk Indonesia

Duta Besar Uni Eropa untuk IndonesiaPetahanaDenis Chaibisejak 2023 Berikut adalah daftar duta besar Uni Eropa untuk Republik Indonesia. Nama Kredensial Selesai tugas Ref. Olof Skoog 30 Januari 2014 [1] Vincent Guerrend 13 November 2015 [2] Vincent Piket 20 November 2019 [3] Denis Chaibi 23 Oktober 2023 Petahana [4] Lihat pula Daftar duta besar untuk Indonesia Hubungan Indonesia dengan Uni Eropa Referensi ^ SBY terima surat kepercayaan dari 13 Dubes LBBP. S...

 

Jackson Sayama

American politician Jackson Doshin SayamaMember of the Hawaii House of Representativesfrom the 21st district20th (2020–2022)IncumbentAssumed office November 3, 2020Preceded byCalvin Say Personal detailsBorn1996 or 1997 (age 26–27)Political partyDemocraticResidence(s)St. Louis Heights, Hawaii[1]Alma materNew York University Shanghai Jackson D. Sayama is an American politician who is currently the Hawaii state representative in Hawaii's 21st district...

 

Mark Aston

English RL coach & former GB international rugby league footballer Mark AstonPersonal informationFull nameMark AstonBorn (1967-09-27) 27 September 1967 (age 56)Castleford, West Riding of Yorkshire, EnglandPlaying informationHeight5 ft 9 in (1.75 m)PositionScrum-half Club Years Team Pld T G FG P 1986–94 Sheffield Eagles 228+4 40 588 34 1370 1987–88 → Bramley (loan) 2 0 0 0 0 1994–95 Featherstone Rovers 34+1 6 85 2 196 1995–03 Sheffield Eagles Total 269 46...

Rotherham Masborough railway station

Disused railway station in South Yorkshire, England This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Rotherham Masborough railway station – news · newspapers · books · scholar · JSTOR (January 2011) (Learn how and when to remove this message) Rotherham MasboroughRotherham Masborough railway station on the las...

 

Serie B 2013-2014

Disambiguazione – Se stai cercando altri significati, vedi Serie B 2013-2014 (disambigua). Serie B 2013-2014Serie B Eurobet 2013-2014 Logo della competizione Competizione Serie B Sport Calcio Edizione 82ª Organizzatore Lega Serie B Date dal 23 agosto 2013al 18 giugno 2014 Luogo  Italia Partecipanti 22 Formula girone unico, play-off e play-out Sito web legaserieb.it Risultati Vincitore Palermo(5º titolo) Altre promozioni EmpoliCesena Retrocessioni NovaraPadovaRegginaJuve Stab...