CiteExport$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_upper_j_idt146",{id:"formSmash:upper:j_idt146",widgetVar:"widget_formSmash_upper_j_idt146",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:upper:exportLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_upper_j_idt147_j_idt149",{id:"formSmash:upper:j_idt147:j_idt149",widgetVar:"widget_formSmash_upper_j_idt147_j_idt149",target:"formSmash:upper:j_idt147:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});

A Tropical Analog of Descartes' Rule of SignsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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Number of Authors: 3
PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2017 (English)In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, no 12, p. 3726-3750Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

2017. no 12, p. 3726-3750
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:su:diva-146006DOI: 10.1093/imrn/rnw118ISI: 000405611600006OAI: oai:DiVA.org:su-146006DiVA: diva2:1136176
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt440",{id:"formSmash:j_idt440",widgetVar:"widget_formSmash_j_idt440",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt446",{id:"formSmash:j_idt446",widgetVar:"widget_formSmash_j_idt446",multiple:true});
Available from: 2017-08-25 Created: 2017-08-25 Last updated: 2017-08-25Bibliographically approved

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.

doi
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