We consider numeration systems where digits are integers and the base is an
algebraic number β such that ∣β∣>1 and β satisfies a
polynomial where one coefficient is dominant in a certain sense. For this class
of bases β, we can find an alphabet of signed-digits on which addition is
realizable by a parallel algorithm in constant time. This algorithm is a kind
of generalization of the one of Avizienis. We also discuss the question of
cardinality of the used alphabet, and we are able to modify our algorithm in
order to work with a smaller alphabet. We then prove that β satisfies
this dominance condition if and only if it has no conjugate of modulus 1. When
the base β is the Golden Mean, we further refine the construction to
obtain a parallel algorithm on the alphabet {−1,0,1}. This alphabet cannot
be reduced any more