Entire functions symmetric in the roots of a polynomial?

Hi,


I consider the function of a complex variable z=x+iy,


f(z) = exp[ w_1(z) ] + exp[ w_2(z) ] + ...+ exp[ w_N(z) ]


where w_n(z) are the N roots of the characteristic polynomial
of the matrix R+zS, with R and S hermitean NxN matrices.


The function f(z) turns out to be entire, since it is symmetric
in the functions w_n(z) which, in turn, are algebraic functions,
being solutions of


g(z,w) = 0,


with g(z,w) a polynomial of order N in z and w.


Much can be said about the function f(z) but I cannot find any
references in the literature, Does anybody have a pointer
to the discussion of this type of entire functions - or just a name
for them?


Any help is appreciated.....................Stefan.Weigert..

.

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