2024/12/23 by Igor Balla, Balla, Igor, Marek Filakovský +7
Engineering · Computer Science · Mathematics · #Advanced Numerical Analysis Techniques #Computational Geometry and Mesh Generation #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2412.18002
Aougab and Gaster [Math. Proc. Cambridge Philos. Soc. 174 (2023), 569-584] proved that any set of simple closed curves on the torus, where any two are non-homotopic and intersect at most k times, has a maximum size of k+O(√(k)log k). We determine the maximum size of such a set for every k. In particular, the maximum never exceeds k+6, and it does not exceed k+4 when k is large. As this quantity coincides with the maximal number of columns of a generic k-modular matrix with two rows, our result also settles the column number problem, a problem of interest in combinatorial optimization, for such matrices.