On certain computations of Pisot numbers

This paper presents two algorithms on certain computations about Pisot numbers. Firstly, we develop an algorithm that finds a Pisot number α   such that Q[α]=FQ[α]=F given a real Galois extension FF of QQ 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 α  , one can compute [αn](modm) in time polynomial in (log(mn))O(1), where m and n are positive integers.

► Two polynomial time algorithms on certain computations about Pisot numbers are proposed.
► The first algorithm is to find a Pisot number generating a given real Galois number field.
► The second algorithm is to compute the modular exponentiation of a Pisot number.