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

We prove the so-called TnTn conjecture: for every real-monic polynomial p(x)p(x) of degree n⩾2n⩾2 there exists an n by n matrix with sign patternTn=-+0⋯0-0⋱⋮0⋱⋱⋱0⋮⋱0+0⋯0-+,whose characteristic polynomial is p(x)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 TnTn and has zeros in positions (1,1)(1,1), and (j+1,j)(j+1,j) for j=2,…,n-1j=2,…,n-1.