Explicit Methods for Radical Function Fields over Finite Fields

TL;DR

This paper presents explicit formulas and algorithms for arithmetic in radical function fields over finite fields, including computations of integral bases, Riemann-Roch spaces, constant fields, genus, and class numbers.

Contribution

It introduces new explicit formulas and algorithms for various arithmetic computations in radical function fields over finite fields, extending previous results and providing practical computational methods.

Findings

Classified places with easy monogenic integral bases.

Provided explicit formulas for functions with specific valuations.

Developed algorithms for computing constant fields, genus, and class numbers.

Abstract

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.