Abstract. We discuss FLINT (Fast Library for Number Theory), a library to support computations in number theory, including highly optimised routines for polynomial arithmetic and linear algebra in exact rings.

We introduce two new packages, Nemo and Hecke, written in the Julia programming language for computer algebra and number theory. We demonstrate that high performance generic algorithms can be implemented in Julia, without the need to resort to a lowlevel C implementation. For specialised algorithms, we use Julia's efficient native C interface to wrap existing C/C++ libraries such as Flint, Arb, Antic and Singular. We give examples of how to use Hecke and Nemo and discuss some algorithms that we have implemented to provide high performance basic arithmetic. ACM Reference format:

Abstract. We investigate two practical divide-and-conquer style algorithms for univariate polynomial arithmetic. First we revisit an algorithm originally described by Brent and Kung for composition of power series, showing that it can be applied practically to composition of polynomials in Z [x] given in the standard monomial basis. We offer a complexity analysis, showing that it is asymptotically fast, avoiding coefficient explosion in Z[x]. Secondly we provide an improvement to Mulders' polynomial division algorithm. We show that it is particularly efficient compared with the multimodular algorithm. The algorithms are straightforward to implement and available in the open source FLINT C library. We offer a practical comparison of our implementations with various computer algebra systems.