A real-rooted polynomial is a nonzero polynomial all of whose complex roots are real
numbers. For a polynomial 
of polynomial
degree 
with nonnegative coefficients, real-rootedness implies
that the sequence 
, ..., 
is an ultra log-concave
sequence, and hence a logarithmically
concave sequence (Xie and Xu 2026). Real-rootedness therefore gives coefficient
inequalities stronger than ordinary log-concavity.
For example, 
is real-rooted and its normalized coefficients 
are all 1. The dependence
polynomial of a graph has a complete real-rootedness
classification due to Xie and Xu (2026).
A nonconstant monic polynomial 
is real-rooted iff it is the characteristic
polynomial 
of a real symmetric matrix 
. Schmeisser (1993, pp. 13-14) gives a tridiagonal
matrix construction from the coefficients, including
repeated roots. For distinct roots,
this matrix is similar
to the companion matrix. Repeated roots
require distinguishing the characteristic
polynomial from the matrix minimal polynomial.
