On alpha-adic expansions in Pisot bases

We study @a-adic expansions of numbers, that is to say, left infinite representations of numbers in the positional numeration system with the base @a, where @a is an algebraic conjugate of a Pisot number @b. Based on a result of Bertrand and Schmidt, we prove that a number belongs to Q(@a) if and only if it has an eventually periodic @a-adic expansion. Then we consider @a-adic expansions of elements of the ring Z[@a^-^1] when @b satisfies the so-called Finiteness property (F). We give two algorithms for computing these expansions - one for positive and one for negative numbers. In the particular case that @b is a quadratic Pisot unit satisfying (F), we inspect the unicity and/or multiplicity of @a-adic expansions of elements of Z[@a^-^1]. We also provide algorithms to generate @a-adic expansions of rational numbers in that case.

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