On the maximal number of columns of a $$\Delta $$-modular integer matrix: bounds and computations
研究了具有m行的Δ-模整数矩阵中两两不同列的最大数量，对固定m给出了关于Δ的上界，并提供了计算方法。
Abstract We study the maximal number of pairwise distinct columns in a $$\varDelta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> </mml:math> -modular integer matrix with m rows. Recent results by Lee et al. provide an asymptotically tight upper bound of $$\mathcal {O}\left( m^2\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msup> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mfenced> </mml:mrow> </mml:math> for fixed $$\varDelta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> </mml:math> . We complement this and obtain an upper bound of the form $$\mathcal {O}(\varDelta )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for fixed m , and with the implied constant depending polynomially on m .