On the Maximal Number of Columns of a $\varDelta $-modular Matrix

Gennadiy Averkov, Matthias Schymura

Abstract: We study the maximal number of pairwise distinct columns in a \(\varDelta \)-modular integer matrix with m rows. Recent results by Lee et al. provide an asymptotically tight upper bound of \(\mathcal {O}(m^2)\) for fixed \(\varDelta \). We complement this and obtain an upper bound of the form \(\mathcal {O}(\varDelta )\) for fixed m, and with the implied constant depending polynomially on m.