2015/10/12 by Jens Forsgård, Dmitry Novikov, Forsgård, J. +3
Computer Science · Mathematics · #26C10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #History and Theory of Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1510.03257
openalex publication_date 2015/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that for any degree d, there exist (families of) finite sequences a0, a1,..., ad 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 a0, a1,... ad respectively. In particular, for any real univariate polynomial P of degree d with non-vanishing constant term, we conjecture that one can take ak = 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.