Almost strict total positivity and a class of Hurwitz polynomials
A real \(n\times n\) matrix \(A:=[a_{ij}]\) is called totally positive (TP) if all its minors are nonnegative and strictly totally positive (STP) if they are positive. A TP matrix \(A\) is called almost strictly totally positive (ASTP) if \(A\) is nonsingular and, for each minor of \(A\) formed by consecutive rows and consecutive columns, the minor is positive if and only if its diagonal entries are positive. Let \(\delta=4.079\dots\) be the real root of \(x^{3}-5x^{2}+4x-1=0\). It has been shown by \textit{T. Craven} and \textit{G. Csordas} [Linear Multilinear Algebra 45, 19--34 (1998; Zbl 0922.15014)] that \(A\) is STP whenever its entries are positive and \(a_{ij}a_{i+1,j+1}\geq\delta a_{i,j+1}a_{i+1,j}\) for all \(i,j=1,\dots,n-1\). The main result of this paper is a generalization of the Craven-Csordas theorem in the case that not all entries of \(A\) are positive. It is known that for any nonsingular TP matrix \(A\) we have: \(a_{ii}>0\) for all \(i\); \(a_{hk}=0\) for all \(h\geq i\) and \(k\leq j\) whenever \(a_{ij}=0\) and \(i>j\); and \(a_{hk}=0\) for all \(h\leq i\) and \(k\geq j\) whenever \(a_{ij}=0\) and \(i<j\). The authors prove that if \(A\) is a nonnegative matrix which satisfies these latter conditions and \[ a_{ij}a_{i+1,j+1}\geq\delta a_{i,j+1}a_{i+1,j} \quad\text{whenever}\quad a_{ij}a_{i+1,j+1}>0, \tag{*} \] then \(A\) is TP\(.\) Moreover if the inequality (*) is always strict, then \(A\) is ASTP. They apply this to show that if the coefficients of \(f(z):=c_{n}z^{n}+c_{n-1}z^{n-1}+\dots+c_{0}\) are positive real numbers and \(c_{k}c_{k+1}\geq\delta c_{k-1}c_{k+1}\) for \(k=1,\dots,n-2\) then \(f(z)\) is Hurwitz stable (that is, all the zeros of \(f(z)\) have negative real parts). They give similar conditions for an entire function to belong to the Laguerre-Pólya class of type I. The authors also show that the \(\delta\) which appears in these theorems cannot be universally replaced by a constant smaller than \(4\), but conjecture that for a fixed value of \(n\) the best constant is \(4\cos^{2}(\pi/(n+1))\).
scientific article; zbMATH DE number 2165383