On the maximal number of columns of a Δ-modular integer matrix: bounds and computations

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 \(\mathcal {O}\left( m^2\right) \) 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.