The Membership Problem for finitely generated quadratic modules in the univariate case

We prove that the Membership Problem is solvable affirmatively for every finitely generated quadratic module Q of R[X1]. For the case that the associated semialgebraic set S is bounded we show that a polynomial f is an element of Q if and only if f is nonnegative on S and fulfills certain order conditions in the boundary points of S. This leads us to the definition of generalized natural generators of the quadratic module Q.