Certificate extraction for rectangular sign patterns

Extend polynomial-time certificate extraction from square sign patterns to rectangular sign patterns with a fixed excess of columns, producing exact numerical witnesses of singularity in the non-sign-nonsingular case.

Background

The paper develops a deterministic polynomial-time algorithm for square sign patterns: it either recognizes sign-nonsingularity or constructs an integer singular realization together with an integer null vector of polynomially bounded bit complexity. The construction relies on extracting opposite-sign determinant terms via even directed cycles, then using coordinatewise affine interpolation to obtain an exactly represented singular matrix.

The authors identify extending this certificate-extraction framework beyond square patterns as the main unresolved direction. In the Consequences section, they specifically note that they make no claim for rectangular L-matrices or positive bounded deficiency and associate that extension with Conjecture 14.12.5 in the Handbook of Satisfiability.