A Tropical Analog of Descartes' Rule of Signs. (27th June 2016) - Forsgård, Jens; Novikov, Dmitry; Shapiro, Boris
Abstract: We prove that for any degree $d$, there exist (families of) finite sequences $\{λ_{k, d}\}_{0\le k\le d}$ of positive numbers such that, for any real polynomial $P$ of degree $d$, the number of its real roots is less than or equal to the number of the so-called essential tropical roots of the polynomial obtained from $P$ by multiplication of its coefficients by $λ_{0, d}, λ_{1, d}, …, λ_{d, d}$, respectively. In particular, for any real univariate polynomial $P(x)$ of degree $d$ with a non-vanishing constant term, we conjecture that one can take $λ_{k, d}={\rm e}^{-k^2}, \, k=0, …, d $ . The latter claim can be thought of as a tropical generalization of Descartes's rule of signs. We settle this conjecture up to degree $4$ as well as a weaker statement for arbitrary real polynomials. Additionally, we describe an application of the latter conjecture to the classical Karlin problem on zero-diminishing sequences.