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a e Q algebraic region algebraic varieties analytic variety Assume f branch of Z(f Cartesian product compact connected compact set concludes the proof connected set countable D-set contained defined dimensional variety domain of holomorphy entire function exactly m zeros exph f is holomorphic f N _ f w1 factor of f finite function f function in f h is entire holomorphic functions homogeneous factor identically zero implies that f LEMMA 15 locally connected modulus space neighborhood of H Newton's identities nonnegative integer number of zeros poly disc polynomial A(w polynomial and h polynomial is homogeneous polynomial of degree power sums PROPOSITION region of type relatively prime residue theorem satisfies the condition satisfies the hypothesis shows slice function symmetric function THEOREM 17 THEOREM 25 thesis vanishes identically w e H wedge zero free zero set zeros of f zeros of g(w