Explicit construction of integral bases of radical function fields

Wu, Qingquan

doi:10.5802/jtnb.714

Cited by 2 publications

(3 citation statements)

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“…Some of these results were already known in special cases; for example, in case D is a squarefree polynomial and gcd(deg D, n) = 1, the function field is superelliptic, and arithmetic in it is described in [GPS02]. Under the assumption that D is a polynomial not divisible by any n-th power, the formula for an integral basis was given in [Wu09]. Our approach generalizes both results.…”

Section: Discussionmentioning

confidence: 66%

“…In this section, we want to generalize a result of Q. Wu [Wu09] on how to give an explicit k[x]-basis of O, the integral closure of k[x] in K. We will need this for computing Riemann-Roch spaces in the next sections.…”

Section: Integral Bases Partmentioning

confidence: 99%

“…In the case of radical function fields over finite constant fields, not much work has been done in the direction of explicit arithmetic. One notable exception is a result by Q. Wu, which gives an explicit k[x]-basis of the integral closure O of k[x] in K [Wu09] under the assumption that D ∈ k[x] is n-th power free. We will reformulate his result in Section 4 to work for all D ∈ k(x).…”

Section: Introductionmentioning

confidence: 99%

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Explicit Methods for Radical Function Fields over Finite Fields

Fontein

2009

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We develop explicit formulas and algorithms for arithmetic in radical function fields K/k(x) over finite constant fields. First, we classify which places of k(x) whose local integral bases have an easy monogenic form, and give explicit formulas for these bases. Then, for a fixed place p of k(x), we give formulas for functions whose valuation is zero for all places P | p except one, for which it is one. We extend a result by Q. Wu on a k[x]-basis of its integral closure in K, show how to compute certain Riemann-Roch spaces and how to compute the exact constant field, resulting in explicit formulas for the exact constant field together with easy to evaluate formulas for the genus of K. Finally, we show how to approximate the Euler product to obtain the class number using ideas of R. Scheidler and A. Stein and give an algorithm. We give bounds on the running time for all algorithms.

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Section: Discussionmentioning

confidence: 66%

Section: Integral Bases Partmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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