Editing Riemann–Roch theorem (section)

{{short description|Relation between genus, degree, and dimension of function spaces over surfaces}}
{{Infobox mathematical statement
| name = Riemann–Roch theorem
| image = 
| caption = 
| field = [[Algebraic geometry]] and [[complex analysis]]
| conjectured by = 
| conjecture date = 
| first proof by = [[Gustav Roch]]
| first proof date = 1865
| open problem = 
| known cases = 
| implied by =
| equivalent to = 
| generalizations = [[Atiyah–Singer index theorem]]<br />[[Grothendieck–Riemann–Roch theorem]]<br />[[Hirzebruch–Riemann–Roch theorem]]<br />[[Riemann–Roch theorem for surfaces]]<br />[[Riemann–Roch-type theorem]]
| consequences = [[Clifford's theorem on special divisors]]<br>[[Riemann–Hurwitz formula]]
}}

The '''Riemann–Roch theorem''' is an important theorem in [[mathematics]], specifically in [[complex analysis]] and [[algebraic geometry]], for the computation of the dimension of the space of [[meromorphic function]]s with prescribed zeros and allowed [[pole (complex analysis)|poles]]. It relates the complex analysis of a connected [[Compact space|compact]] [[Riemann surface]] with the surface's purely topological [[genus (mathematics)|genus]] ''g'', in a way that can be carried over into purely algebraic settings.

Initially proved as '''Riemann's inequality''' by {{harvtxt|Riemann|1857}}, the theorem reached its definitive form for Riemann surfaces after work of [[Bernhard Riemann|Riemann]]'s short-lived student {{harvs|txt|authorlink=Gustav Roch|first=Gustav|last=Roch|year=1865}}. It was later generalized to [[algebraic curve]]s, to higher-dimensional [[algebraic variety|varieties]] and beyond.