{
  "id": "math/0302276",
  "version": "v5",
  "published": "2003-02-22T08:59:15.000Z",
  "updated": "2006-10-02T18:46:18.000Z",
  "title": "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)",
  "authors": [
    "Apoloniusz Tyszka"
  ],
  "comment": "12 pages, LaTeX2e, the version that appeared in Journal of Geometry",
  "journal": "Journal of Geometry 85 (2006), no. 1-2, pp. 188-199",
  "categories": [
    "math.MG"
  ],
  "abstract": "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).",
  "revisions": [
    {
      "version": "v5",
      "updated": "2006-10-02T18:46:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51M05"
    ],
    "keywords": [
      "commutative field extending",
      "preserves unit distance preserves",
      "beckman-quarles theorem",
      "discrete form",
      "finite set"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......2276T"
    }
  }
}