A Tropical Analog of Descartes' Rule of Signs. (27th June 2016)

• Record Type: Journal Article
• Authors: Forsgård, Jens; Novikov, Dmitry; Shapiro, Boris
• Is Part Of: International mathematics research notices. Volume 2017:Number 12(2017)
• Page Start: 3726
• Page End: 3750
• Publication Date: 2016-06-27
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnw118
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: 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.