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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Mnëv's universality theorem is a result in the intersection of
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
and
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
used to represent algebraic (or semialgebraic) varieties as realization spaces of oriented matroids. Informally it can also be understood as the statement that point configurations of a fixed combinatorics can show arbitrarily complicated behavior. The precise statement is as follows: : Let V be a semialgebraic variety in ^n defined over the integers. Then V is stably equivalent to the realization space of some oriented matroid. The theorem was discovered by Nikolai Mnëv in his 1986 Ph.D. thesis.


Oriented matroids

For the purposes of this article, an ''oriented matroid'' of a finite subset S\subset ^n is the list of partitions of S induced by hyperplanes in ^n (each oriented hyperplane partitions S into the points on the "positive side" of the hyperplane, the points on the "negative side" of the hyperplane, and the points that lie on the hyperplane). In particular, an oriented matroid contains the full information of the incidence relations in S, inducing on S a
matroid In combinatorics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid Axiomatic system, axiomatically, the most significant being in terms ...
structure. The realization space of an oriented matroid is the space of all configurations of points S\subset ^n inducing the same oriented matroid structure.


Stable equivalence of semialgebraic sets

For the purpose of this article stable equivalence of semialgebraic sets is defined as described below. Let U and V be semialgebraic sets, obtained as a disjoint union of connected semialgebraic sets :U=U_1\coprod \cdots\coprod U_k\, and \,V=V_1\coprod \cdots\coprod V_k We say that U and V are ''rationally equivalent'' if there exist homeomorphisms \phi_i:U_i \to V_i defined by rational maps. Let U\subset ^, V\subset ^ be semialgebraic sets, :U=U_1\coprod \cdots\coprod U_k\, and \,V=V_1\coprod \cdots\coprod V_k with U_i mapping to V_i under the natural projection \pi deleting the last d coordinates. We say that \pi: U \to V is a ''stable projection'' if there exist integer polynomial maps \varphi_1, \ldots, \varphi_\ell, \psi_1, \dots, \psi_m:\; ^n \to ^d such that U_i =\. The ''stable equivalence'' is an equivalence relation on semialgebraic subsets generated by stable projections and rational equivalence.


Implications

Mnëv's universality theorem has numerous applications in algebraic geometry, due to
Laurent Lafforgue Laurent Lafforgue (; born 6 November 1966) is a French mathematician. He has made outstanding contributions to Langlands' program in the fields of number theory and Mathematical analysis, analysis, and in particular proved the Langlands conjecture ...
,
Ravi Vakil Ravi D. Vakil (born February 22, 1970) is a Canadian-American mathematician working in algebraic geometry. He is the current president of the American Mathematical Society. Education and career Vakil attended high school at Martingrove Collegiat ...
and others, allowing one to construct moduli spaces with arbitrarily bad behaviour. This theorem together with Kempe's universality theorem has been used also by Kapovich and Millson in the study of the moduli spaces of linkages and arrangements. Mnëv's universality theorem also gives rise to the ''universality theorem for
convex polytopes ''Convex Polytopes'' is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional polyhedron, convex polyhedra. It was written by Branko Grünbaum, with contributions from Victor Klee, M ...
''. In this form it states that every semialgebraic set is stably equivalent to the realization space of some convex polytope. Jürgen Richter-Gebert showed that universality already applies to polytopes of dimension four.


References


Further reading

* ''
Convex Polytopes ''Convex Polytopes'' is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional polyhedron, convex polyhedra. It was written by Branko Grünbaum, with contributions from Victor Klee, M ...
'' by
Branko Grünbaum Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentReal algebraic geometry Oriented matroids Theorems in algebraic geometry Theorems in combinatorics