Maximal and linearly inextensible polynomials

Details Maximal and linearly inextensible polynomials Maximal and linearly inextensible polynomials

2006-01-25

arxiv.org/abs/math/0601600v3

Mathematica Scandinavica vol 99:1 (2006), 53-75.

Mathematics

Complex Variables

Final version, to appear in Mathematica Scandinavica, 16 pages, no figures, LaTeX2e

Scientific paper

Let S(n,0) be the set of monic complex polynomials of degree $n\ge 2$ having all their zeros in the closed unit disk and vanishing at 0. For $p\in S(n,0)$ denote by $|p|_{0}$ the distance from the origin to the zero set of $p'$. We determine all 0-maximal polynomials of degree $n$, that is, all polynomials $p\in S(n,0)$ such that $|p|_{0}\ge |q|_{0}$ for any $q\in S(n,0)$. Using a second order variational method we then show that although some of these polynomials are linearly inextensible, they are not locally maximal for Sendov's conjecture.

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