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 consider the Bipartite Perfect Matching function, which is the indicator over all bipartite graphs having a perfect matching, and show that its approximate degree is $\widetilde{\Theta}(n^{1.5})$. The upper bound is obtained by fully characterizing the unique multilinear polynomial representing the Boolean dual of the perfect matching function, over the Reals. In particular, 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 $\Theta(n^{1.5})$.

翻译：Boolean 函数的近似度为 $\ ell\ infty $- norm 的纯多线性多线性多元接近度 。 我们考虑的是Boolean 超立方体上真实多线性多线性多元接近度的最小度 。 我们考虑的是双线性完美匹配函数, 这是所有双面图中具有完美匹配性的指标, 并显示其大约度是$\ lobletilde { theta} (n\log n) 。 上界是通过充分描述代表 Expercube\ cite {aron202020 度和 Huang2019 的完美匹配功能的布尔性双倍值来获得的。 我们显示, 极性匹配的光谱性敏感度非常小 $\ $\ 15} 我们显示, 完美匹配的光谱性是 $\ {n} 。