Curves testing boundedness of polynomials on subsets of the real plane
Let S?R2 S ? R 2 be a semialgebraic set. We exhibit a family of semialgebraic plane curves ?c ? c , c?0 c ? 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 degf . More precisely, each ?c ? c is a sum of at most l  continuous semialgebraic curves ? i c , each parametrized by a Puiseux polynomial, where the number l  and the family of curves ? i c 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.
semialgebraic sets bounded polynomials puiseux series