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Prime divisors hyperelliptic curves

  1. Sep 15, 2011 #1
    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):

    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] )
    Last edited by a moderator: Apr 26, 2017
  2. jcsd
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