Smooth ideals in hyperelliptic function fields

Smooth Ideals in Hyperelliptic Function Fields Smooth Ideals in Hyperelliptic Function Fields

Recently, several algorithms have been suggested for solving the discrete logarithm problem in the Jacobians of high-genus hyperelliptic curves over nite elds. Some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are suuciently dense. We explicitly show how these density results can be derived. All proofs are purely combinatorial and do not ...

Smooth ideals in hyperelliptic function fields

Recently, several algorithms have been suggested for solving the discrete logarithm problem in the Jacobians of high-genus hyperelliptic curves over finite fields. Some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are sufficiently dense. We explicitly show how these density results can be derived. All proofs are purely combinatorial and d...

Uniform Approximation of Abhyankar Valuation Ideals in Smooth Function Fields

Let R be an n-dimensional regular local domain essentially of finite type over a ground field k of characteristic zero, and let ν be a rank one valuation centered on R. Recall that this is equivalent to asking that ν be an R-valued valuation on the fraction field K of R, taking non-negative values on R and positive values on the maximal ideal m ⊆ R. A theorem of Zariski and Abhyankar states tha...

Reduced Ideals in Function Fields

Let F denote a function eld of transcendence degree one over a nite eld k. We assume that the eld is tamely rami ed at in nity, that the valuations at in nity of a set of fundamental units are known and we have gcd(f1; : : : ; fs) = 1, where fi denotes the degree of a place at in nity. In such a situation we describe a simple arithmetic in the divisor class group. One draw back of this arithmet...

Automorphism groups of hyperelliptic function fields