Theorem of Riemann-Roch

The theorem of Riemann-Roch is a result of geometry. Originally, he answers the problem to seek if there exist rational functions on a surface of Riemann $S$ given, having with poles of multiplicity imposed in certain points. For example, in its weak form, the theorem states that for $m$ points given, the space of the rational functions on $S$ with more the one first order pole in these points and having finished elsewhere is of size finished on $C$ larger than $m-g+1$, where $g$ is the Genre surface.

More precisely, that is to say $D=\; (a\_v)$ ($v$ describing the valuations on the body of the rational functions of surface $S$, or an equivalent way points of $S$) an unspecified divider and $\backslash \; Delta$ a canonical divider (i.e. associated with a differential form). If one calls $l\; (D)$ the dimension of the vector space formed of the rational functions on surface such as $v\; (F)\; \backslash \; geq\; -\; a\_v$ for all $v\; \backslash \; in\; S$, one a:

Theorem of Riemann-Roch :

$l\; (D)\; -\; L\; (\backslash \; Delta-D)\; =deg\; (D)+1-g.$

This theorem can be inteprété like a calculation of characteristic of Euler-Poincaré for this situation. There are many demonstrations and generalizations.

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