Recognizing the maximum finite number of solutions of an absolute value equation

For rational $A$ and $b$, the paper gives a polynomial-time test for whether
$Ax + |x| = b$ has exactly $2^n$ distinct solutions, using $n + 1$ rational
linear-programming feasibility problems. No nonsingularity or spectral-radius
promise is required, and infinite solution sets are rejected. The proof
characterizes the maximum finite solution count by the geometry of a polytope
with paired inequalities: there are exactly $2^n$ solutions if and only if
$b > 0$, the polytope $P = \{y : (A + I)y \le b,\ (A - I)y \le b\}$ is
bounded, and $b - Ay$ is entrywise positive on $P$. Necessity follows from
Cramer's rule and multiaffine interpolation of the determinants of
$I + A\operatorname{diag}(d)$ over $d \in [-1, 1]^n$. For sufficiency, every vertex of $P$ is a solution, and the
sign patterns of the vertices are closed under flipping any one coordinate.