  We obtain complete characterizations of the Unique Bipartite Perfect Matching
function, and of its Boolean dual, using multilinear polynomials over the
reals. Building on previous results, we show that, surprisingly, the dual
description is sparse and has low $\ell_1$-norm -- only exponential in
$\Theta(n \log n)$, and this result extends even to other families of
matching-related functions. Our approach relies on the M\"obius numbers in the
matching-covered lattice, and a key ingredient in our proof is M\"obius'
inversion formula.
  These polynomial representations yield complexity-theoretic results. For
instance, we show that unique bipartite matching is evasive for classical
decision trees, and nearly evasive even for generalized query models. We also
obtain a tight $\Theta(n \log n)$ bound on the log-rank of the associated
two-party communication task.
