In search of Newton-type inequalities (Q6539285)

The authors consider univariate polynomials with positive coefficients: \(Q:=\sum _{j=0}^na_jx^j\), \(a_j>0\). They present new results which give a relationship between the double ratios \(q_j:=a_j^2/a_{j-1}a_{j+1}\) and the number of real roots of the polynomial \(Q\). In particular, if \(n\geq 4\) and the number of real roots of \(Q\) is \(\geq n-2\), then \(q_1+q_{n-1}\geq 2A\), \(A:=2(n^2-3n)/(n-2)^2\). Hence, either \(q_1\geq A\) or \(q_{n-1}\geq A\). The polynomial \((x+1)^{n-2}(x^2+((n-4)/(n-2))x+1)\), with \(q_1=q_{n-1}=A\), has exactly \(n-2\) real roots, so the above estimate is sharp. The authors also provide counterexamples to two earlier conjectures refining Descartes' rule of signs.