  The approximate degree of a Boolean function is the least degree of a real
multilinear polynomial approximating it in the $\ell_\infty$-norm over the
Boolean hypercube. We show that the approximate degree of the Bipartite Perfect
Matching function, which is the indicator over all bipartite graphs having a
perfect matching, is $\widetilde{\Theta}(n^{3/2})$.
  The upper bound is obtained by fully characterizing the unique multilinear
polynomial representing the Boolean dual of the perfect matching function, over
the reals. Crucially, we show that this polynomial has very small $\ell_1$-norm
-- only exponential in $\Theta(n \log n)$. The lower bound follows by bounding
the spectral sensitivity of the perfect matching function, which is the
spectral radius of its cut-graph on the hypercube \cite{aaronson2020degree,
huang2019induced}. We show that the spectral sensitivity of perfect matching is
exactly $\Theta(n^{3/2})$.
