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, p. 3). 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 finite simple graph
has a complete real-rootedness classification due to Xie and Xu (2026, p. 4).
The derivative of a nonconstant real-rooted polynomial is again real-rooted by Rolle's theorem. Repeated differentiation therefore preserves real-rootedness until a nonzero constant is reached.
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.
A monic polynomial of fixed positive polynomial degree in one variable
whose coefficients are real
analytic in one real parameter 
on an open interval, and
which is real-rooted for every 
in that interval, admits real
analytic root functions (Alekseevsky et al. 1998).
These functions need not remain in increasing order. For example, 
has the analytic roots 
and 
,
whereas the increasing choice 
, 
is not differentiable at 
.
For smooth coefficients, root functions can always be chosen twice differentiable,
but their second derivatives need not be continuous
(Kriegl et al. 2004). Thus twice differentiable
does not mean C2. The restriction
to one parameter is essential for the real
analytic statement. The polynomial 
is real-rooted for every 
, but its roots
cannot be chosen differentiably near 
(Alekseevsky et al. 1998).
