On mappings of the Galois space (Q1076970)

Let V be an n-dimensional metric vector space over a field K, \(\delta\) : \(V\times V\to K\) a distance function. Let \(ind V\) be the Witt index, that is, the dimension of all maximal isotropic subspaces, of V. Given a map \(\sigma\) : \(V\to V\) such that \(\delta (p,q)=1\) implies \(\delta (p^{\sigma},q^{\sigma})=1\), if \(n=2\), \(ind V=1\) and \(char K\neq 2,3,5,\) then \(\sigma\) has been proven to be semilinear [\textit{W. Benz}, J. Geom. 17, 193-201 (1981; Zbl 0499.51005), and J. Geom. 18, 70-77 (1982; Zbl 0499.51006); the author, J. Geom. 21, 164-183 (1983; Zbl 0547.51005)]. If \(n\geq 3\), \(K={\mathbb{R}}\), and \(\delta\) is either euclidean or minkowskian (that is, the bilinear form related to \(\delta\) has signature n or n-2, respectively), then \(\sigma\) is known to be linear [\textit{W. Benz}, Arch. Math. 34, 550-559 (1980; Zbl 0446.51015); \textit{J. Lester}, C. R. Math. Acad. Sci., Soc. R. Can. 3, 59-61 (1981; Zbl 0474.51024)]. The author now proves: If \(K=GF(p^ m)\), \(p>2\) and \(n\geq 3\), then \(\sigma\) is semilinear up to a translation, provided \(n\not\equiv 0,-1,-2(mod p)\) or the discriminant of V satisfies a certain condition. The proof is based on the conditions for regular simplexes to exist in a Galois space.