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

TL;DR

This paper derives generalized inequalities related to Hermite's theorem, providing conditions for real zeros of polynomials through matrix minors and graph analysis, extending classical discriminant concepts.

Contribution

It introduces a new matrix $E(n)$ and generalizes discriminant expressions, linking minors of Hermite matrices to these new constructs for real polynomial zeros.

Findings

Expressions for Hermite inequalities are established.

The $(k+1)$-th minor relates to the $k$-th minor of $E(n)$ matrix.

Conditions for positive zeros are characterized by functions $M(m_2,m_1,n)$.

Abstract

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.