Prime divisors hyperelliptic curves

[yavanna]

Every divisor [itex]D[/itex] associated to a hyperelliptic curve over [itex]\mathbb{F}_q[/itex] can be represented by a couple of polynomials [itex]D=div(a(x),b(x))[/itex] (Mumfor representation):
http://www.google.it/url?url=http:/...pyTszxEIfDtAa5yaGDBw&ved=0CCMQygQwAA&cad=rja"

A divisor [itex]D[/itex] is prime if [itex]a(x)[/itex] is irreducible over [itex]\mathbb{F}_q[/itex].
I'm trying to prove that a semi-reduced divisor can be written as a combination of prime divisors factoring [itex]a(x) [/itex]:
if [itex]a(x)=\prod a_{i}(x)^{c_i} [/itex], then [itex]D=\sum c_i div(a_i(x),y-b_i(x))[/itex], where [itex]b_i \equiv b \textit{mod}a_i[/itex]

I've some problems with [itex]y-b_i(x)[/itex], any ideas?

(I think the author of the article had some notation problems, if we express [itex]D=div(a(x),b(x))[/itex] it should be [itex]D=\sum c_i div(a_i(x),b_i(x))[/itex], where [itex]D=div(a(x),b(x))[/itex] means [itex]D=gcd (div(a(x)),div(y-b(x)))[/itex] )

 

[jvicens]

Thank you for your post. The Mumford representation of a divisor D=div(a(x),b(x)) on a hyperelliptic curve over \mathbb{F}_q is a useful tool for studying the properties of divisors on such curves. In order to address your question about the semi-reduced divisor, I would first like to clarify a few things about the Mumford representation.

The Mumford representation of a divisor D=div(a(x),b(x)) is a way of writing D as a sum of prime divisors. In this representation, the prime divisors are of the form div(a_i(x),y-b_i(x)), where a_i(x) and b_i(x) are polynomials and y is a variable. The key property of this representation is that the divisors in the sum are pairwise relatively prime, meaning that any two of them have no common factors.

Now, let's consider your question about a semi-reduced divisor. A semi-reduced divisor is a divisor D=div(a(x),b(x)) where a(x) and b(x) have no common factors, but b(x) may have repeated factors. In this case, we can write b(x)=\prod b_i(x)^{c_i}, where the b_i(x) are pairwise relatively prime and c_i are positive integers. Using this factorization, we can rewrite D as a sum of prime divisors:

D=\sum c_i div(a(x),y-b_i(x))

Note that this is similar to the Mumford representation, except that the b_i(x) are not necessarily linear in y. However, we can still use this representation to study the properties of semi-reduced divisors.

Now, let's address your question about the notation y-b_i(x). This notation is used to indicate that the divisor div(a_i(x),y-b_i(x)) is a prime divisor, meaning that a_i(x) is irreducible over \mathbb{F}_q. This is important because it allows us to factor a semi-reduced divisor into a sum of prime divisors, as shown above.

I hope this helps clarify the Mumford representation and how it relates to semi-reduced divisors. If you have further questions or need more clarification, please let me know. Thank you for your interest in this topic.