Iterated rings of bounded elements and generalizations of Schmüdgen's Positivstellensatz

Let A be a commutative R–algebra of finite transcendence degree d ∈ N. We investigate the relationship between the subring of (geometrically) bounded elements H(A) := {a ∈ A | ∃ν ∈ N : |a| ≤ ν on SperA} and the subring of arithmetically bounded elements H (A) := {a ∈ A | ∃ν ∈ N : ν + a and ν − a are sums of squares in A}. Obviously, H ′(A) ⊆ H(A). In 1991, Schmüdgen proved the remarkable theorem that A = H(A) implies A = H ′(A) if A is finitely generated. In 1996, Becker and Powers considered the chain A ⊇ H(A) ⊇ H(H(A)) =: H(A) ⊇ . . . and showed H(A) = H(A). In 1998, Monnier related both results and conjectured H(A) = H ′(A) which generalizes both of them at the same time. We prove this conjecture and develop tools to study H ′(A). One of the applications is the following: If a ∈ A is “small at infinity” and a ≥ 0 on SperA, then a+ ε is a sum of squares in A for every ε > 0.