Fast computation of integral bases

TL;DR

This paper presents new complexity bounds for efficiently computing integral bases in number and function fields, achieving near-linear cost in certain cases and advancing the speed of related algebraic computations.

Contribution

It introduces improved complexity bounds and algorithms for computing integral bases, including fractional ideals, using recent algorithmic advances and precise analysis.

Findings

Achieved softly linear complexity for function fields with large residual characteristic.

Extended results to fractional ideals for faster Riemann-Roch space computation.

Provided detailed complexity analysis based on recent algorithms.

Abstract

We obtain new complexity bounds for computing a triangular integral basis of a number field or a function field. We reach for function fields a softly linear cost with respect to the size of the output when the residual characteristic is zero or big enough. Analogous results are obtained for integral basis of fractional ideals, key ingredients towards fast computation of Riemann-Roch spaces. The proof is based on the recent fast OM algorithm of the authors and on the MaxMin algorithm of Stainsby, together with optimal truncation bounds and a precise complexity analysis.