On Totally Positive Functions, LaPlace Integrals and Entire Functions of the LaGuerre-Polya-Schur Type.

We say K is consistent at p if each r(u, p) lies in V(K, p) in a one-one manner. This is obviously a necessary condition that K be smoothly imbeddable in Em. It may happen that a relation a1v, + ... + akvk = 0 is not true at p, yet arbitrarily near p a relation arbitrarily near this one holds. This might preclude smooth imbedding. A necessary and sufficient condition for imbeddability of a general complifold seems highly difficult to obtain. Instead, we shall give a slight restriction on K. 4. Celiwise homogeneity. Let K be consistent at each p. If for each a, and cells ai., ox,, with a as face, the part of V(K, p) over these cells is of constant dimension as p moves over a, we say K is celiwise homogeneous. THEOREM: Any celiwise homogeneous complifold of dimension n may be smoothly imbedded in E2n+1, and smoothly immersed in E2n.