ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

4. Algebraic Geometry

1991 MS Classification: not given Schulte, Udo, Technical University Ilmenau, Germany From a difference equation for integers to a special closed curve for each positive integer The recurrence relation x_{n+k}^{(k)} = x_{n}^{(k)} of integers modulo k > 1 with initial conditions x_{l}^{(k)} = l , 0 ≤ l < k , l ∈ N_{0}, has the solution

x_{n}^{(k)} = \frac{k − 1}{2} + ∑_{j=1}^{k−1} \frac{e^{\frac{2πijn}{k}}}{e^{− \frac{2πij}{k}} − 1} .

Extending n to a real variable x the functions f_{k}(x) can be defined. The parametric curves (Re(f_{k}(x)) , Im(f_{k}(x))) are closed curves with positive curvature meeting the points {0, 1, …, k − 1} on a straight line.