Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A discrete form of the Beckman-Quarles theorem for mappings from R^2
  (C^2) to F^2, where F is a subfield of a commutative field extending R (C)   Let F be a subfield of a commutative field extending R. Let phi_n:F^n \times
F^n ->F, phi_n((x_1,...,x_n),(y_1,...,y_n))=(x_1-y_1)^2+...+(x_n-y_n)^2. We say
that f:R^n->F^n preserves distance d>=0 if for each x,y \in R^n |x-y|=d implies
phi_n(f(x),f(y))=d^2. Let A_n(F) denote the set of all positive numbers d such
that any map f:R^n->F^n that preserves unit distance preserves also distance d.
Let D_n(F) denote the set of all positive numbers d with the property: if x,y
\in R^n and |x-y|=d then there exists a finite set S(x,y) with {x,y} \subseteq
S(x,y) \subseteq R^n such that any map f:S(x,y)->F^n that preserves unit
distance preserves also the distance between x and y. Obviously, {1} \subseteq
D_n(F) \subseteq A_n(F). We prove: A_n(C) \subseteq {d>0: d^2 \in Q} \subseteq
D_2(F).
  Let K be a subfield of a commutative field Gamma extending C. Let psi_2:
Gamma^2 \times Gamma^2->Gamma,
psi_2((x_1,x_2),(y_1,y_2))=(x_1-y_1)^2+(x_2-y_2)^2. We say that f:C^2->K^2
preserves unit distance if for each X,Y \in C^2 psi_2(X,Y)=1 implies
psi_2(f(X),f(Y))=1. We prove: if X,Y \in C^2, psi_2(X,Y) \in Q and X \neq Y,
then there exists a finite set S(X,Y) with {X,Y} \subseteq S(X,Y) \subseteq C^2
such that any map f:S(X,Y)->K^2 that preserves unit distance satisfies
psi_2(X,Y)=psi_2(f(X),f(Y)) and f(X) \neq f(Y).

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The categories of this paper are: hep-ph
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The categories of this paper are: q-alg
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
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The categories of this paper are: adap-org
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The categories of this paper are: atom-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: solv-int
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The categories of this paper are: ao-sci
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The categories of this paper are: supr-con
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The categories of this paper are: stat
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The categories of this paper are: nlin
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The categories of this paper are: nucl-th
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The categories of this paper are: math-ph