Fix a semialgebraic set S in \(\mathbb {R}^n\) and a function \(g:S\rightarrow \mathbb {R}\) . We address the question on how to determine the power m such that for a polynomial f the function \({|f| }/{|g|^m}\) is bounded on S. As a consequence, we give explicit formulae for growth rate of a polynomial, i.e., the optimal power m when \(g=\Vert x\Vert \) , on a certain type of sets, called weighted tentacles, in terms of their support. To this aim, we study algebras of bounded polynomials for these sets and give an interpretation of the results in terms of convex conical hulls of integer points. In particular, we apply the results to constructively describe growth rates of polynomials on any semialgebraic subset of the real plane. Moreover, we give the monomial generators for the algebra of bounded polynomials on basic semialgebraic set described by quasi-homogeneous inequalities.
