Negative inertia of entrywise powers

For $\alpha>0$, let $M_n(\alpha)$ be the largest possible number ofnegative eigenvalues of $A^{\circ\alpha}$, where $A$ ranges over the realpositive definite matrices of order $n$ with strictly positive entries.For $n\geq3$, we prove\[ M_n(\alpha)=n-2 \qquad\left(0<\alpha<\frac1{n-2}\right),\]and, for every fixed $\alpha>0$,\[ \lim_{n\to\infty}\frac{M_n(\alpha)}n = \begin{cases} 0,&\alpha\in\mathbb N,\\ 1,&\alpha\notin\mathbb N. \end{cases}\]For a positive integer $m$, put\[ p_m(n)=\min\left\{p\geq1:n\leq\binom{p+m-1}{m}\right\}.\]For $m\geq2$, we prove\[ M_n(\alpha)\geq n-p_m(n) \qquad\left(\frac1m\leq\alpha<\frac1{m-1}\right),\]and hence $n-M_n(\alpha)=O_\alpha(n^{1/m})$ on this interval.At reciprocal exponents we further prove that, for $m\geq2$, $p\geq3$,and $1\leq r\leq\lfloor p/3\rfloor$,\[ M_{\binom{p+m-1}{m}+r}(1/m) \geq \binom{p+m-1}{m}+r-p.\]Finally, we characterize the equality $M_{m+2}(1/m)=m-1$ by a reversetensor inequality.