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A Tropical Analog of Descartes' Rule of Signs
Stockholm University, Faculty of Science, Department of Mathematics.
Number of Authors: 3
2017 (English)In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, no 12, 3726-3750 p.Article in journal (Refereed) Published
Abstract [en]

We prove that for any degree d, there exist (families of) finite sequences {lambda(k,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 lambda(0,d),lambda(1,d),..,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) = 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.

Place, publisher, year, edition, pages
2017. no 12, 3726-3750 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-146006DOI: 10.1093/imrn/rnw118ISI: 000405611600006OAI: oai:DiVA.org:su-146006DiVA: diva2:1136176
Available from: 2017-08-25 Created: 2017-08-25 Last updated: 2017-08-25Bibliographically approved

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