A tropical analog of Descartes' rule of signs

J. Forsgård; D. Novikov; B. Shapiro

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arXiv · 1510.03257

A tropical analog of Descartes' rule of signs

J. Forsgård·D. Novikov·B. Shapiro

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Abstract

We prove that for any degree d, there exist (families of) finite sequences a_0, a_1,..., a_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 a_0, a_1,... a_d respectively. In particular, for any real univariate polynomial P of degree d with non-vanishing constant term, we conjecture that one can take a_k = 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.

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BibTeX RIS

J. Forsgård, D. Novikov, B. Shapiro. 2015-10-12. A tropical analog of Descartes' rule of signs. https://arxiv.org/abs/1510.03257

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