A Real Holomorphy Ring without the Schmüdgen Property

Abstract

AbstractA preordering T is constructed in the polynomial ring A = ℝ[t1, t2, … ] (countablymany variables) with the following two properties: (1) For each f ∈ A there exists an integer N such that −N ≤ f (P) ≤ N holds for all P ∈ SperT(A)). (2) For all f ∈ A, if N + f, N − f ∈ T for some integer N, then f ∈ R. This is in sharp contrast with the Schmüdgen-Wörmann result that for any preordering T in a finitely generated ℝ-algebra A, if property (1) holds, then for any f ∈ A, f > 0 on SperT(A)) f ∈; T. Also, adjoining to A the square roots of the generators of T yields a larger ring C with these same two properties but with ∑C2 (the set of sums of squares) as the preordering.