  We prove expressions for the inequalities in Hermite's theorem which are
conditions for a real polynomial to have real zeros. These expressions
generalize the discriminant of a quadratic polynomial and the expression of J.
Mar\'ik for a cubic polynomial. We show that the $(k+1)$-th minor of the
Hermite matrix associated a polynomial $p(x)$ is equal to the $k$-th minor of
another matrix we call $E(n)$ times $n^{k-1}$ and a simple integer. To prove
this equivalence, we prove generalizations of the discriminant of a polynomial
and analyze certain labeled directed graphs. To define this matrix $E(n)$ we
define functions $M(m_2,m_1,n)$ which are positive if the zeros of $p(x)$ are
positive.
