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The pp conjecture for spaces of orderings of rational conics. (English) Zbl 1113.11024
The space of orderings \((X,G)\) is completely determined by the group structure and the ternary relation \(a\in D(b,c)\) on \(G.\) A positive-primitive (pp for short) formula is a formula of the type \(\exists_{v_1,\dots,v_n} \psi(v_1,\dots,v_n,w_1,\dots,w_k)\), where \(\psi(v_1,\dots,v_n,w_1,\dots,w_k)\) is a finite conjuction of atoms of the form \(1\in D(a\prod v_i^{\varepsilon_i}, b\prod v_i^{\delta_i}),\) where \(\varepsilon_i, \delta_i\in \{0,1\}\) and \(a, b\) being products of \(\pm 1\) and a finite number of \(w_j\)’s. The second author [J. Symb. Logic 67, 341–352 (2002; Zbl 1018.11017)] stated the following Open Problem: Is it true that every pp formula which holds on every finite subspace of a space of orderings holds on the whole space? It is known that this abstract “local-global principle” holds for typical formulas as “two forms are isometric”, “an element is represented by a form”, “a form is isotropic”. The answer to the Open Problem is “yes” for many classes of spaces of orderings [see J. Symb. Log. 71, No. 4, 1097–1107 (2006; Zbl 1129.12005)] and the paper mentioned above. The authors investigate the space of orderings of the function field of a real irreducible conic over the field of rational numbers. They construct conterexamples showing that the pp conjecture fails for the space of orderings of the function field of an ellipse without rational points and also in the case of two parallel lines.

11E10 Forms over real fields
12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
Full Text: DOI
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