Choosing roots of polynomials with symmetries smoothly

Details Choosing roots of polynomials with symmetries smoothly Choosing roots of polynomials with symmetries smoothly

2006-03-28

arxiv.org/abs/math/0603660v1

Rev. Mat. Complut. 20 (2007), No. 2, 267--291.

Mathematics

Classical Analysis and ODEs

19 pages, 2 figures, LaTeX

Scientific paper

The roots of a smooth curve of hyperbolic polynomials may not in general be parameterized smoothly, even not $C^{1,\alpha}$ for any $\alpha > 0$. A sufficient condition for the existence of a smooth parameterization is that no two of the increasingly ordered continuous roots meet of infinite order. We give refined sufficient conditions for smooth solvability if the polynomials have certain symmetries. In general a $C^{3n}$ curve of hyperbolic polynomials of degree $n$ admits twice differentiable parameterizations of its roots. If the polynomials have certain symmetries we are able to weaken the assumptions in that statement.

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