Harnack's curve theorem

"M-curve" redirects here. For a curve describing the demographics of working women, see <a href="/facts/Career_woman/YGED6KMa">Career woman § Limited work options</a>.

In <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, Harnack's curve theorem, named after <a href="/facts/Carl_Gustav_Axel_Harnack/p7BYQstF">Axel Harnack</a>, gives the possible numbers of <a href="/facts/Connected_space/5CUTxfoA">connected components</a> that an <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curve</a> can have, in terms of the degree of the curve. For any algebraic curve of degree m in the real <a href="/facts/Projective_plane/p6ot8oNv">projective plane</a>, the number of components c is bounded by

1
              −
              (
              −
              1
              
                )
                
                  m
                
              
            
            2
          
        
        ≤
        c
        ≤
        
          
            
              (
              m
              −
              1
              )
              (
              m
              −
              2
              )
            
            2
          
        
        +
        1.
         
      
    
    {\displaystyle {\frac {1-(-1)^{m}}{2}}\leq c\leq {\frac {(m-1)(m-2)}{2}}+1.\ }

The maximum number is one more than the maximum <a href="/facts/Geometric_genus/S2oQVNCb">genus</a> of a curve of degree m, attained when the curve is <a href="/facts/Singularity_theory/MgLkyrhl">nonsingular</a>. Moreover, any number of components in this range of possible values can be attained.

A curve which attains the maximum number of real components is called an M-curve (from "maximum") – for example, an <a href="/facts/Elliptic_curve/60jvaHSA">elliptic curve</a> with two components, such as 
 
 
 
 
 y
 
 2
 
 
 =
 
 x
 
 3
 
 
 −
 x
 ,
 
 
 {\displaystyle y^{2}=x^{3}-x,}
 
 or the <a href="/facts/Trott_curve/BGVJkJ13">Trott curve</a>, a <a href="/facts/Quartic_plane_curve/5j35LcSS">quartic</a> with four components, are examples of M-curves.
This theorem formed the background to <a href="/facts/Hilbert%2527s_sixteenth_problem/SUR6JBKB">Hilbert's sixteenth problem</a>.
In a recent development a Harnack curve is shown to be a curve whose <a href="/facts/Amoeba_(mathematics)/DaS8wBFu">amoeba</a> has area equal to the <a href="/facts/Newton_polygon/CBtfnV8X">Newton polygon</a> of the <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> P, which is called the characteristic curve of <a href="/facts/Dimer_model/blw6bzyy">dimer models</a>, and every Harnack curve is the spectral curve of some dimer model.(Mikhalkin 2001)(Kenyon, Okounkov & Sheffield (2006))

<ul><li><a href="/facts/Dmitry_Gudkov_(mathematician)/RhR1TlNY">Dmitrii Andreevich Gudkov</a>, 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</li>
<li><a href="/facts/Carl_Gustav_Axel_Harnack/p7BYQstF">Carl Gustav Axel Harnack</a>, <a href="https://link.springer.com/article/10.1007%2FBF01442458">Ueber die Vieltheiligkeit der ebenen algebraischen Curven</a>, Math. Ann. 10 (1876), 189–199</li>
<li>George Wilson, Hilbert's sixteenth problem, <a href="/facts/Topology_(journal)/D2ejns02">Topology</a> 17 (1978), 53–74</li>
<li><a href="/facts/Richard_Kenyon/RYIK2Wka">Kenyon, Richard</a>; <a href="/facts/Andrei_Okounkov/ME33ML8t">Okounkov, Andrei</a>; <a href="/facts/Scott_Sheffield/uofP8St7">Sheffield, Scott</a> (2006). "Dimers and Amoebae". <a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a>. 163 (3): 1019–1056. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math-ph/0311005">math-ph/0311005</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4007%2Fannals.2006.163.1019">10.4007/annals.2006.163.1019</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2215138">2215138</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:119724053">119724053</a>.</li>
<li>Mikhalkin, Grigory (2001), Amoebas of algebraic varieties, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0108225">math/0108225</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2001math......8225M">2001math......8225M</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2102998">2102998</a></li></ul>

Harnack's curve theorem open-in-new

Harnack's curve theorem