{
  "id": "math/0404321",
  "version": "v6",
  "published": "2004-04-19T19:16:00.000Z",
  "updated": "2004-11-10T15:25:37.000Z",
  "title": "The Beckman-Quarles theorem for mappings from C^2 to C^2",
  "authors": [
    "Apoloniusz Tyszka"
  ],
  "comment": "9 pages, LaTeX2e, a new proof uses the connectivity of unit-distance graph on C^2",
  "journal": "Acta Math. Acad. Paedagog. Nyh\\'azi. (N.S.) 21 (2005), no. 1, 63-69 (electronic), www.emis.de/journals",
  "categories": [
    "math.MG"
  ],
  "abstract": "Let phi: C^2 times C^2 -> C, phi((x_1,x_2),(y_1,y_2))=(x_1-y_1)^2+(x_2-y_2)^2. We say that f:C^2->C^2 preserves unit distance, if for each X,Y in C^2 phi(X,Y)=1 implies phi(f(X),f(Y))=1. We prove that each unit-distance preserving mapping f:C^2->C^2 has a form J circ (gamma,gamma), where gamma:C->C is a field homomorphism and J:C^2->C^2 is an affine mapping with orthogonal linear part. We also prove a more general result for any commutative field K for which {x \\in K: x^2+1=0} \\neq \\emptyset and char(K) \\not\\in {2,3,5}.",
  "revisions": [
    {
      "version": "v6",
      "updated": "2004-11-10T15:25:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "39B32",
      "51B20",
      "51M05"
    ],
    "keywords": [
      "beckman-quarles theorem",
      "orthogonal linear part",
      "preserves unit distance",
      "general result",
      "field homomorphism"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......4321T"
    }
  }
}