User:Catalina marina

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Coordinate ring[edit]

The coordinate ring of C over K is defined as

.


The polynomial is irreducible over , so

is an integral domain.
Proof. If r (x,y) were reducible over , it would factor as (y - u(x)) · (y - v(x)) for some u,v. But then u(x) · v(x)= f(x) so it has degree 2g + 1, and a(x) + b(x) = h(x) so it has degree smaller than g, which is impossible.


Note that any polynomial function can be written uniquely as

  with ,


Norm and degree[edit]

The conjugate of a polynomial function G(x,y) = u(x) - v(x)y in is defined to be

.

The norm of G is the polynomial function . Note that N(G) = u(x)2 + u(x)v(x)h(x) - v(x)2f(x), so N(G) is a polynomial in only one variable.


If G(x,y) = u(x) - v(x) · y, then the degree of G is defined as

.

Properties:


The function field K(C) of C over K is the field of fractions of K[C], and the function field of C over is the field of fractions of . The elements of are called rational functions on C. For R such a rational function, and P a finite point on C, R is said to be defined at P if there exist polynomial functions G, H such that R = G/H and H(P) ≠ 0, and then the value of R at P is

.

For P a point on C that is not finite, i.e. P = , we define R(P) as:

If   then .
If   then   is not defined.
If   then   is the ratio of the leading coefficients of G and H.


For and ,

If then R is said to have a zero at P,
If R is not defined at P then R is said to have a pole at P, and we write .


Order of a polynomial function at a point[edit]

For and , the order of G at P is defined as:

if P = (a,b) is a finite point which is not Weierstrass. Here r is the highest power of (x-a) which divides both u(x) and v(x). Write G(x,y) = (x - a)r(u0(x) - v0(x)y) and if u0(a) - v0(a)b = 0, then s is the highest power of (x - a) which divides N(u0(x) - v0(x)y = u02 + u0v0h - v02f, otherwise, s = 0.
if P = (a,b) is a finite Weierstrass point, with r and s as above.
if P = O.