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We prove expressions for the inequalities in Hermite's theorem which areconditions for a real polynomial to have real zeros. These expressionsgeneralize 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 theHermite matrix associated a polynomial $p(x)$ is equal to the $k$-th minor ofanother matrix we call $E(n)$ times $n^{k-1}$ and a simple integer. To provethis equivalence, we prove generalizations of the discriminant of a polynomialand analyze certain labeled directed graphs. To define this matrix $E(n)$ wedefine functions $M(m_2,m_1,n)$ which are positive if the zeros of $p(x)$ arepositive.

On the inequalities in Hermite's theorem for a real polynomial to have real zeros