---
abstract: 'We describe how to solve simultaneous [Padé]{}approximations over a power series ring ${{\mathsf{K}}}[[x]]$ for a field ${{\mathsf{K}}}$ using ${O^{\scriptscriptstyle \sim}\!}(n^{\omega - 1} d)$ operations in ${{\mathsf{K}}}$, where $d$ is the sought precision and $n$ is the number of power series to approximate. We develop two algorithms using different approaches. Both algorithms return a reduced sub-bases that generates the complete set of solutions to the input approximations problem that satisfy the given degree constraints. Our results are made possible by recent breakthroughs in fast computations of minimal approximant bases and Hermite [Padé]{}approximations.'
author:
- |
    Johan Rosenkilde, né Nielsen\
    Technical University of Denmark\
    Denmark\
    jsrn@jsrn.dk
- |
    Arne Storjohann\
    University of Waterloo\
    Canada\
    astorjoh@uwaterloo.ca
- |
    Johan Rosenkilde, né Nielsen\
    \
    \
    Arne Storjohann\
    \
    \
bibliography:
- 'bibtex.bib'
title: 'Algorithms for Simultaneous [Padé]{}Approximations [^1] '
---

Introduction {#sec:intro}
============

The Simultaneous [Padé]{}approximation problem concerns approximating several power series $S_1,\ldots,S_n \in {{\mathsf{K}}}[[x]]$ with rational functions $\frac {\sigma_1}\lambda,\ldots,\frac {\sigma_n} \lambda$, all sharing the same denominator $\lambda$. In other words, for some $d \in {\mathbb Z\xspace}_{\geq 0}$, we seek $\lambda \in {{\mathsf{K}}}[x]$ of low degree such that each of $${{\textnormal{rem}}}(\lambda S_1,\  x^d) , {{\textnormal{rem}}}(\lambda S_2,\  x^d),\ \ldots,\ {{\textnormal{rem}}}(\lambda S_n,\  x^d)$$ has 