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DTSTART;TZID=America/Los_Angeles:20200123T121500
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DTSTAMP:20260409T004413
CREATED:20191008T203742Z
LAST-MODIFIED:20200120T195443Z
UID:1599-1579781700-1579785000@colleges.claremont.edu
SUMMARY:Dragging the roots of a polynomial to the unit circle (Sinai Robins\, University of Sao Paulo)
DESCRIPTION:Several conditions are known for a self-inversive polynomial that ascertain the location of its roots\, and we present a framework for comparison of those conditions. We associate a parametric family of polynomials p_α(x) to each such polynomial p\, and define cn(p)\, il(p) to be the sharp threshold values of α that guarantee that\, for all larger values of the parameter\, p_α(x) has\, respectively\, all roots in the unit circle and all roots interlacing the roots of unity of the same degree.  Interlacing implies circle rootedness\, hence il(p) ≥ cn(p)\, and this inequality is often used for showing circle rootedness. Both il(p) and cn(p) turn out to be semi-algebraic functions of the coefficients of p\, and some useful bounds are also presented\, entailing several known results about roots in the circle. The study of il(p) leads to a rich classification of real self-inversive polynomials of each degree\, organizing them into a complete polyhedral fan. We have a close look at the class of polynomials for which il(p) = cn(p)\, whereas in general the quotient il(p)/cn(p) is shown to be unbounded as the degree grows. Several examples and open questions are presented.  This is joint work with Arnaldo Mandel.
URL:https://colleges.claremont.edu/ccms/event/antc-seminar-sinai-robins-university-of-sao-paulo/
LOCATION:Emmy Noether Room\, Millikan 1021\, Pomona College\, 610 N. College Ave.\, Claremont\, California\, 91711
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
GEO:34.099908;-117.7142522
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