Maximal and linearly inextensible polynomials - Borcea, Julius (MATHEMATICA SCANDINAVICA, 2006)
Let S(n,0) be the set of monic complex polynomials of degree n2 having all their zeros in the closed unit disk and vanishing at 0. For pS(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 pS(n,0) such that |p|_0|q|_0 for any qS(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.