On Certain Computations of Pisot Numbers

TL;DR

This paper introduces two algorithms related to Pisot numbers: one for finding a Pisot number with a given Galois extension, and another for efficiently computing powers modulo m.

Contribution

It provides a polynomial-time lattice reduction algorithm for constructing Pisot numbers from Galois extensions and an efficient method for modular exponentiation of Pisot numbers.

Findings

Algorithm for constructing Pisot numbers from Galois extensions in polynomial time.

Efficient computation of $ ext{[}\alpha^n ext{]} mod m$ in polynomial time.

Demonstrates practical algorithms for algebraic number computations involving Pisot numbers.

Abstract

This paper presents two algorithms on certain computations about Pisot numbers. Firstly, we develop an algorithm that finds a Pisot number $\alpha$ such that $\Q[\alpha] = \F$ given a real Galois extension $\F$ of $\Q$ by its integral basis. This algorithm is based on the lattice reduction, and it runs in time polynomial in the size of the integral basis. Next, we show that for a fixed Pisot number $\alpha$, one can compute $ [\alpha^n] \pmod{m}$ in time polynomial in $(\log (m n))^{O(1)}$, where $m$ and $n$ are positive integers.