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

Abstract

We prove that the Membership Problem is solvable affirmatively for every finitely generated quadratic module Q of R[X-1]. 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. (C) 2012 Elsevier B.V. All rights reserved.