Abstract:  Let  f  be a Morse-Bott function on a compact oriented finite dimensional manifold M.  The polynomial Morse inequalities and a perturbation technique approximating any Morse-Bott function by a Morse function show immediately that if  MB_t(f)  is the Morse-Bott polynomial of  f , and  P_t(M) is the Poincare polynomial of  M , there exists a polynomial  R(t)  such that  MB_t(f) = P_t(M) + (1+t)R(t).  We prove that  R(t)  is a polynomial with non-negative integer coefficients, using the Morse Homology Theorem, and by studying the ranks of the kernels of the differentials of the Morse-Smale-Witten complexes of the Morse functions involved in the approximation scheme.
Our method works when all the critical submanifolds are oriented or when Z_2  coefficients are used.  Our proof is much more  direct than all previous proofs.
This is a joint work with David  E. Hurtubise.