Descartes' rule of signs is exact!
Descartes' rule of signs states that for any non-zero polynomial \(f\in\mathbb{R}[x]\) with \(t\) positive roots counted with multiplicities, the difference between the number of sign changes in the vector of coefficients of \(f\), denoted by \(S(f)\), and \(t\) is a non-negative even integer. Poincaré showed that this rule can be made exact, i.e., when the two quantities match up completely, by premultiplying \(f\) by a correction factor that depends on \(f\). The author first sharpens Poincaré's result and shows that the correction factor can be chosen as a polynomial with non-negative coefficients. As a second main result, the following general statement is proven: For any fixed \(f\in\mathbb{R}[x]\), the sequence \(S((x+1)^n f(x))\), \(n=0,1,\dots\) is monotone decreasing with limit equal to the number of positive roots of \(f\) counted with multiplicities. The paper is very well written and has a nice historical introduction to the topic.
scientific article; zbMATH DE number 5840425