Small Matrices with Large Inverses: Unimodular $4\times 4$ Cases

Abstract

How close to singularity can an $n\times n$ unimodular matrix be? For ternary cases as $n$ increases, exact expressions are unlikely, but upon fixing $n=4$ and assessing $(2k+1)$-ary cases as $k$ increases, we make significant progress; similarly for $(k+1)$-ary cases of $4\times 4$ nonnegative unimodular matrices.

arXiv comment21 pages; $5\times 5$ conjectures appear at end; $6\times 6$ and $7\times 7$ signed conjectures likewise