On the maximal number of columns of a $$\Delta $$-modular integer matrix: bounds and computations

Abstract

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