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Choosing roots of polynomials with symmetries smoothly
The roots of a smooth curve of hyperbolic polynomials may not in general be
parameterized smoothly, even not for any . 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 curve of hyperbolic polynomials
of degree admits twice differentiable parameterizations of its roots. If
the polynomials have certain symmetries we are able to weaken the assumptions
in that statement.Comment: 19 pages, 2 figures, LaTe
Symmetry algebra for the generic superintegrable system on the sphere
The goal of the present paper is to provide a detailed study of irreducible
representations of the algebra generated by the symmetries of the generic
quantum superintegrable system on the -sphere. Appropriately normalized, the
symmetry operators preserve the space of polynomials. Under mild conditions on
the free parameters, maximal abelian subalgebras of the symmetry algebra,
generated by Jucys-Murphy elements, have unique common eigenfunctions
consisting of families of Jacobi polynomials in variables. We describe the
action of the symmetries on the basis of Jacobi polynomials in terms of
multivariable Racah operators, and combine this with different embeddings of
symmetry algebras of lower dimensions to prove that the representations
restricted on the space of polynomials of a fixed total degree are irreducible
Normal Form, Symmetry and Infinite Dimensional Lie Algebra for System of Ode's
The normal form for a system of ode's is constructed from its polynomial
symmetries of the linear part of the system, which is assumed to be
semi-simple. The symmetries are shown to have a simple structure such as
invariant function times symmetries of degree one called basic symmetries. We
also show that the set of symmetries naturally forms an infinite dimensional
Lie algebra graded by the degree of invariant polynomials. This implies that if
this algebra is non-commutative then the method of multiple scales with more
than two scaling variables fails to apply.Comment: 10 page
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