The results which we present here form the part of guiding by A.M.

Principal circle bundle over a PL polyhedron can be triangulated and thus obtains combinatorics. The triangulation is assembled from triangulated circle bundles over simplices. To every triangulated circle bundle over a simplex we associate a necklace (in combinatorial sense). We express rational local formulas for all powers of first Chern class in the terms of mathematical expectations of parities of the associated necklaces. This rational parity is a combinatorial isomorphism invariant of triangulated circle bundle over simplex, measuring mixing by triangulation of the circular graphs over vertices of the simplex. The goal of this note is to sketch the logic of deduction these formulas from Kontsevitch's cyclic invariant connection form on metric polygons.

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. IntroductionLet R be any real closed field. A set of sign conditions C = [C,, C> , C>] is a list of three finite sets of polynomials in Z[X1, . . . , Xk]. The basic semialgebraic set defined by C, S(C) C Rk, is the setWe shall prove a completeness theorem.We associate to the set of sign conditions C (i) a geometric object: the ring Q(C) of quadratic functions over S(C) equipped (ii) a syntactic object, R(C), obtained as a quotient of the inductive limit of provableWe will then prove the following theorem (Theorem 7.1): The ring Q(C) of quadratic functions is isomorphic to the ring R(C).Next we prove the following (Theorem 7.2):sets C 1 and C2 of sign conditions are quadmtic-ring equivalent (the rings Q(Cl) and Q(C2) are isomorphic) if an only if they are logically equivalent, that U, if they can be deduced from each other by wing simple deduction rules. with its sets of nonnegative and positive functions; ronsequences of C under some simple deduction rules. 1991 hfoihemaiicr Subjeci Cladrificoiion. Keywords and phrwcs. 246 Math. Nachr. 181 (1!)1)11)The key to these results is the link between algebra and geometry provided Iiy Stengle's positivstellensatz ([S], [BCR]). The next step in the future development of this theory (which originates from t1wideas of the second author [MI) will consist in defining algebraic topological invarit1iit.n associated to nets of small sign conditions which are stable under the simple deducticw rules. The fact that only small sign conditions are used, and that the simple deducticw rules are local and can be performed by transforming a fixed number of small siKii conditions without looking at the rest of the net, will be essential in the developrrrcwt of the algebraic topological aspects of the theory. As a consequence, if two sets (11 small sign conditions have different invariants, then the corresponding rings arc tic11 isomomorphic. Quadratic functions associated to a basic semialgebraic* set Definition 2.1. The ring Q(C) is the ring generated by the coordinate functiorih XI , . . . , Xk and finite compositions of the following operations: (i) additions and multiplications, (ii) inversion of a function never vanishing within S(C), (iii) extraction of the nonnegative square root of nonnegative function on S(C).We immediately obtain the following properties:(iv) f E Q(C) is nonnegative on S(C) if and only if it is a square in Q(C), (v) f E Q(C) is nonzero on S(C) if and only if it is invertible in Q(C).So the subset of nonnegative functions and the subset of nonzero functions can tiv described via the ring structure of Q(C).The ring of quadratic functions has nice algebraic properties since its elements ciiii be described by finite algebraic information, but it also contains information about t1iv geometry of the original basic semi-algebraic set. For example we can prove (usiiibi, the notations in [BCR]):Proposition 2.2. The canonical morphism from Z[X1,. . . , X k ] to Q(C) induces ( I homeomorphism between the real spectrum of the ring Q(C) and the constructible w /