Nonnegative Morse polynomial functions and polynomial optimization

In this paper we study the representation of Morse polynomial functions which are nonnegative on a compact basic closed semi-algebraic set in \(\mathbb R^{n}\), and having only finitely many zeros in this set. Following Bivià-Ausina (Math Z 257:745–767, 2007), we introduce two classes of non-degenerate polynomials for which the algebraic sets defined by them are compact. As a consequence, we study the representation of nonnegative Morse polynomials on these kinds of non-degenerate algebraic sets. Moreover, we apply these results to study the polynomial optimization problem for Morse polynomial functions.