REALIZABILITY OF COMBINATORIAL TYPES OF CONVEX POLYHEDRA OVER FIELDS N. E. Mnev UDC 513.34 It is shown that the minimal subfield of the field of real numbers over which all real combinatorial types of convex polyhedra can be realized is the field of all real algebraic numbers. Let 5 = [~o,~41 be a block-scheme, So be a finite set, SI be a system of subsets of So ; C a realization over ~ of the block-scheme S (where F is an ordered field) i.e., a polytope (convex polyhedron) in the space F ~, such that the block-scheme formed by the set of its vertices and system of hyperfaces, as a system of subsets of the set of vertices, is isomorphic with S . A configuration ~ in the projective plane ~ is called an ~ -realization of the block-scheme S over the projective plane ~ , if the block- scheme formed by the points of ~ and the lines of ~ , as subsets of the set of points, is isomorphic with S . The block-scheme of a polytope (configuration) will be called its com- binatorial type. Polytopes (configurations) with given combinatorial type are called combin- atorially equivalent. Some block-schemes are not combinatorial types. A given combinatorial type can have a C - (or ~ -) realization over one field and not over another. M. Perles con- structed a block-scheme of twelve points having a G-realization over R (for I=8 ) and not having a C -realization over ~ (cf. [1]). In other words, he gave an example of a polytope with twelve vertices in ~$, whose vertices cannot possibly be placed simultaneous- ly at rational points (i.e., be realized in ~8 ) so as to preserve the combinatorial type. A. M. Vershik suggested to the author proving the following improvement of this result: THEOREM. The minimal subfield of the field ~ over which all combinatorial types of polytopes realizable over ~ are realizable is the field of all real algebraic numbers ~ . In other words, i) for any finite extension ~ of the field of rational numbers, there ex- ists a polytope in ~, whose combinatorial type is not realizable over ~ ; 2) any com- binatorial type of polytopes realizable over ~ is realizable over ~ . In its own right, the basic auxiliary assertion is Lemma 1 which was suggested by Vershik and which generalizes Perles' construction and refines the basic theorem of projective geome- try. In Sec. i ~ we construct a series of block-schemes with the following property: they are -realizable over ~ , but for each finite extension ~D ~ one can find a block-scheme in this series which is not ~ -realizable over ~. In Sec. 2, with the help of the Gale transformation we make the transition from configurations to polytopes, which gives a series of block-schemes with the same property but not for G -realizability. In Sec. 3 we prove Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 123, pp. 203-207, 1983. 606 0090-4104/85/2804- 0606509.50 9 1985 Plenum Publishing Corporation