Consider the problem of optimizing a polynomial f(X) for X=(x1,''' , xn) in a closed rectangular-form domain D of the real Euclidean space Rn(X) under constraint polynomial equations HS=0, where HS = {h1,''', hr} R[X]. Applying some modified form of Zero Decomposition Theorem to the HS we get in an algorithmic way a finite set of real values K= Keg(f, D, HS) such that the least or greatest value of K is just the least or greatest value of f to be determined. Numerous problems involving inequalities can be settled by means of the above finiteness theorem.