Real Algebraic and Analytic Geometry
Submission: 2009, May 22.
We prove that the Membership Problem is solvable affirmatively for every finitely generated quadratic module Q of IR[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 Q and an algorithm which produces at most three generators of Q.
Mathematics Subject Classification (2000): 12E05, 12L12, 12Y05, 14P10.
Keywords and Phrases: quadratic modules, membership problem, positive polynomials and sums of squares, definability.
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