Curves testing boundedness of polynomials on subsets of the real plane

Let S⊂R2S⊂R2 be a semialgebraic set. We exhibit a family of semialgebraic plane curves ΓcΓc, c⩾0c⩾0, such that a polynomial f∈R[X,Y]f∈R[X,Y] is bounded on S if and only if it is bounded on a finite number of curves from this family. This number depends on S and deg f  . More precisely, each ΓcΓc is a sum of at most l   continuous semialgebraic curves Γic, each parametrized by a Puiseux polynomial, where the number l   and the family of curves Γic depend on the set S only. To this aim we describe the algebras of bounded polynomials on tentacles of the set S which determine the algebra of polynomials bounded on S.