A Tropical Analog of Descartes’ Rule of Signs
Jens Forsgård, Dmitry Novikov, Boris Zalmanovich Shapiro · International Mathematics Research Notices · 2016
We prove that for any degree |$d$|, there exist (families of) finite sequences |$\{\lambda_{k,d}\}_{0\le k\le d}$| 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 |$\lambda_{0,d},\lambda_{1,d},\dots , \lambda_{d,d}$|, 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 |$\lambda_{k,d}={\rm e}^{-k^2},\,k=0,\dots, 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.