Algebra of Polynomials Bounded on a Semi-algebraic Set f≤r

The algebra of polynomials in ℝ [ x ] which are bounded on a semi-algebraic set determined by a polynomial inequality f ( x ) ≤ r with f (0) = 0 is studied and the case when it is generated by a finite set of monomials is discussed. A large class of polynomials which are asymptotic to finitely many monomials (including nondegenerate polynomials) is introduced and the algebra of polynomials bounded on [ f ≤ r ] can be determined by a cone and is independent on r > 0, where f belongs to this class. Note that the set of all polynomials whose supports lie in a given closed convex cone in the first quadrant forms an algebra generated by a finite set of monomials. In other cases, we can give upper and lower bounds of the algebra via outer normal cones of the faces of the Newton polyhedron. As a consequence, some sufficient conditions which ensure that the algebra under consideration is generated by finitely many monomials is given.