A Tropical Analog of Descartes’ Rule of Signs.

We prove that for any degree d, there exist (families of) finite sequences {λk,d}0≤k≤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 λ0,d,λ1,d,…,λd,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 λk,d=e−k2, 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. [ABSTRACT FROM AUTHOR]