First counterexamples are given to a basic question raised in [10]. The paper considers the space of orderings (X,G) of the function field of a real irreducible conic [Formula: see text] over the field ℚ of rational numbers. It is shown that the pp conjecture fails to hold for such a space of orderings when [Formula: see text] has no rational points. In this case, it is shown that the pp conjecture "almost holds" in the sense that, if a pp formula holds on each finite subspace of (X,G), then it holds on each proper subspace of (X,G). For pp formulas which are product-free and 1-related, the pp conjecture is known to be true, at least if the stability index is finite [11]. The counterexamples constructed here are the simplest sort of pp formulas which are not product-free and 1-related.

Abstract. Multirings are objects like rings but with multi-valued addition. They are a variant of other objects called hyperrings, defined by Krasner [12], [13]. In [16] the second author defines multirings, introduces a certain special class of multirings called real reduced multirings, defines a natural reflection A Q red (A) from the category of multirings satisfying −1 / ∈ A 2 to the full subcategory of real reduced multirings, provides an elementary first-order description of these objects, and proves that these objects are precisely the spaces of signs, also known as abstract real spectra, considered earlier in [1], [15]. In the present paper we extend results of E. Becker and others concerning orderings of higher level on fields and rings to orderings of higher level on hyperfields and multirings and, in the process of doing this, we establish higher level analogs of the results in [16]. In particular, we introduce a class of multirings called -real reduced multirings, define a natural reflection A Q -red (A) from the category of multirings satisfying −1 / ∈ A 2 to the full subcategory of -real reduced multirings, and provide an elementary first-order description of these objects. The relationship between -real reduced hyperfields and the spaces of signatures defined by Mulcahy and Powers [20], [21], [22] is also examined.

Abstract. Witt equivalent fields can be understood to be fields having the same symmetric bilinear form theory. Witt equivalence of finite fields, local fields and global fields is well understood. Witt equivalence of function fields of curves defined over archimedean local fields is also well understood. In the present paper, Witt equivalence of general function fields over global fields is studied. It is proved that for any two such fields K, L, any Witt equivalence K ∼ L induces a cannonical bijection v ↔ w between Abhyankar valuations v on K having residue field not finite of characteristic 2 and Abhyankar valuations w on L having residue field not finite of characteristic 2. The main tool used in the proof is a method for constructing valuations due to Arason, Elman and Jacob [1]. The method of proof does not extend to non-Abhyankar valuations. The result is applied to study Witt equivalence of function fields over number fields. It is proved, for example, that if k, ℓ are number fields and k(x 1 , . . . , xn) ∼ ℓ(x 1 , . . . , xn), n ≥ 1, then k ∼ ℓ and the 2-ranks of the ideal class groups of k and ℓ are equal.