Gebze Technical University General Seminars
Evolution of roots of iterated derivatives of polynomials
Mohan Ravichandran
MSGSU, Turkey
Özet :
The Gauss-Lucas theorem says that the convex hull of the
roots of the derivative of a polynomial is contained in the convex
hull of the roots of the polynomial. Qualitatively, the convex hulls
of the roots shrink under taking derivative. Can one make this
quantitative? Given a polynomial $p$ , let $\sigma(p)$ denote the convex hull of the roots of $p$. A few weeks ago, I proved the following theorem that to my surprise turns out to be new. For any degree $n$ polynomial $p$ and any $ c \geq \dfrac{1}{2}$,
\[\operatorname{Area} \sigma (p^{(cn)}) \leq 4(c-c^2) \operatorname{Area}\sigma(p).\]
Here $p^{(cn)}$ is shorthand for the $\lfloor cn \rfloor$'th derivative of $p$. This constant is independent of the polynomial $p$ or
even the degree $n$.
Interestingly, the proof of the theorem is not particularly hard - The
ingredients include a remarkable technique due to Joshua Batson, Adam
Marcus, Daniel Spielman and Nikhil Srivastava which they call the
barrier method and an easy to state and prove but powerful theorem
concerning the relation between the roots of a polynomial and its
derivative due to Rajesh Pereira and independently, Semyon Malamud.
I'll give the proof of this quantitative Gauss-Lucas theorem and talk
about the connections to geometric functional analysis. I'll also
discuss a version of this result for random polynomials and tell you
about some very interesting(at least to me) numerical simulations that
I have made around this problem, which throw up plenty of questions.
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Tarih |
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11.11.2016 |
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Saat |
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14:00 |
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Yer |
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Gebze Teknik Üniversitesi, Matematik Bölümü Seminer Odası |
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Dil |
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English |
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Ek Dosya |
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Özet |