A proof of the Tn conjecture: Centralizers,Jacobians and spectrally arbitrary sign patterns

We prove the so-called $T_n$ conjecture: for every real-monic polynomial $p(x)$ of degree $n?2$ there exists an n by n matrix with sign pattern$T_n=\left[\begin{array}{ccccc}\hfill -\hfill & \hfill +\hfill & \hfill 0\hfill & \hfill ?\hfill & \hfill 0\hfill \\ \hfill -\hfill & \hfill 0\hfill & \hfill ?\hfill & \hfill \hfill & \hfill ?\hfill \\ \hfill 0\hfill & \hfill ?\hfill & \hfill ?\hfill & \hfill ?\hfill & \hfill 0\hfill \\ \hfill ?\hfill & \hfill \hfill & \hfill ?\hfill & \hfill 0\hfill & \hfill +\hfill \\ \hfill 0\hfill & \hfill ?\hfill & \hfill 0\hfill & \hfill -\hfill & \hfill +\hfill \end{array}\right]\text{,}$whose characteristic polynomial is $p(x)$. The proof converts the problem of determining the nonsingularity of a certain Jacobi matrix to the problem of proving the non-existence of a nonzero matrix B that commutes with a nilpotent matrix with sign pattern $T_n$ and has zeros in positions $(1\text{,}1)$, and $(j+1\text{,}j)$ for $j=2\text{,}\dots \text{,}n-1$.