Parallel and online arithmetics in imaginary quadratic fields

Nestandardní číselné systémy jsou určené svou bází p é C, p > 1, a svou abecedou cifer A c C. Zabýváme se polygonálními číselnými systémy s abecedou ve tvaru A„= (0, 1, p,..., p" ), kde P = e ~ . Navíc požadujeme, aby báze i abeceda byly v okruhu celých čísel nějakého imaginárního kva-Non-standard numeration systems are given by their base P é C, P > 1, and their alphabet of digits A c C. We focus on the so-called polygonal numeration systems where the alphabet is of the form A„= (0, 1, P,..., P ') where P = e ~ and both the base and the alphabet are in the ring of algebraic integers of some imaginary quadratic field. Feasibility of several arithmetic operations including parallel addition and on-line division and multiplication is discussed. We characterize the complete polygonal numeration systems in imaginary quadratic fields. The Extending Window Method [20] is used to find the algorithms for parallel addition. Then the decision whether the numeration systems satisfy OL property follows along with computation of preprocessing for on-line division using the implementation from [29]