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}_0kd 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=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.