In mathematics, Mnëv's universality theorem is a result in the intersection of combinatorics and algebraic geometry 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 {\displaystyle V} be a semialgebraic variety in R n {\displaystyle {\mathbb {R} }^{n}} defined over the integers. Then V {\displaystyle 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 ⊂ R n {\displaystyle S\subset {\mathbb {R} }^{n}} is the list of partitions of S {\displaystyle S} induced by hyperplanes in R n {\displaystyle {\mathbb {R} }^{n}} (each oriented hyperplane partitions S {\displaystyle 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 {\displaystyle S} , inducing on S {\displaystyle S} a matroid structure. The realization space of an oriented matroid is the space of all configurations of points S ⊂ R n {\displaystyle S\subset {\mathbb {R} }^{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 {\displaystyle U} and V {\displaystyle V} be semialgebraic sets, obtained as a disjoint union of connected semialgebraic sets
U = U 1 ∐ ⋯ ∐ U k {\displaystyle U=U_{1}\coprod \cdots \coprod U_{k}\,} and V = V 1 ∐ ⋯ ∐ V k {\displaystyle \,V=V_{1}\coprod \cdots \coprod V_{k}}
We say that U {\displaystyle U} and V {\displaystyle V} are rationally equivalent if there exist homeomorphisms ϕ i : U i → V i {\displaystyle \phi _{i}:U_{i}\to V_{i}} defined by rational maps. Let U ⊂ R n + d , V ⊂ R n {\displaystyle U\subset {\mathbb {R} }^{n+d},V\subset {\mathbb {R} }^{n}} be semialgebraic sets,
U = U 1 ∐ ⋯ ∐ U k {\displaystyle U=U_{1}\coprod \cdots \coprod U_{k}\,} and V = V 1 ∐ ⋯ ∐ V k {\displaystyle \,V=V_{1}\coprod \cdots \coprod V_{k}}
with U i {\displaystyle U_{i}} mapping to V i {\displaystyle V_{i}} under the natural projection π {\displaystyle \pi } deleting the last d {\displaystyle d} coordinates. We say that π : U → V {\displaystyle \pi :U\to V} is a stable projection if there exist integer polynomial maps
φ 1 , … , φ ℓ , ψ 1 , … , ψ m : R n → R d {\displaystyle \varphi _{1},\ldots ,\varphi _{\ell },\psi _{1},\dots ,\psi _{m}:\;{\mathbb {R} }^{n}\to {\mathbb {R} }^{d}}
such that
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