Analysis of logarithmic factors in column bases complexity

Determine the exact asymptotic number of logarithmic factors in the arithmetic complexity of algorithms for computing column bases of polynomial matrices (specifically the algorithm of Zhou and Labahn, 2013) as used within HermiteDiagonal for extracting diagonal entries of the Hermite normal form, and replace the placeholder constant c1 with a rigorously derived bound.

Background

In the complexity analysis of HermiteDiagonal, the authors rely on kernel bases and column bases subroutines for polynomial matrices. While the kernel bases cost includes analyzed logarithmic factors, the column bases cost lacks a precise analysis of its logarithmic components.

To proceed, the paper introduces a constant c1 to capture the unknown logarithmic factors, explicitly noting that this aspect requires thorough analysis.