A rational set in the plane is a point set with all its pairwise distances rational. Ulam and Erdós conjectured in 1945 that there is no dense rational set in the plane. In this paper we associate special surfaces, we call them distance surfaces, to finite (rational) sets in the plane. We prove that under a mild condition on the points in the rational set S the associated distance surface in P 3 is a surface of general type. Also if we assume Bombieri-Lang Conjecture in arithmetic algebraic geometry we can answer the Erdós-Ulam problem.

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Motivated by Schur’s result on computing the Galois groups of the exponential Taylor polynomials, this paper aims to compute the Galois groups of the Taylor polynomials of the elementary functions [Formula: see text] and [Formula: see text]. We first show that the Galois groups of the [Formula: see text]th Taylor polynomials of [Formula: see text] are as large as possible, namely, [Formula: see text] (full symmetric group) or [Formula: see text] (alternating group), depending on the residue of the integer number [Formula: see text] modulo [Formula: see text]. We then compute the Galois groups of the [Formula: see text]th Taylor polynomials of [Formula: see text] and show that these Galois groups essentially coincide with the Coexter groups of type [Formula: see text] (or an index 2 subgroup of the corresponding Coexter group).

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Let X ⊂ R n be a bounded Lipschitz domain and consider the energy functional Fσ 2 [u; X] := X F(∇u) dx, over the space of admissible maps Aϕ(X) := {u ∈ W 1,4 (X, R n) : det ∇u > 0 for L n-a.e. in X, u| ∂X = ϕ}, where the integrand F : Mn×n → R is quasiconvex and sufficiently regular. Here our attention is paid to the prototypical case when F(ξ) := 1 2 σ2(ξ)+Φ(det ξ). The aim of this paper is to discuss the question of multiplicity versus uniqueness for extremals and strong local minimizers of Fσ 2 and the relation it bares to the domain topology. In contrast, for constructing explicitly and directly solutions to the system of Euler-Lagrange equations associated to Fσ 2 , we use a topological class of maps referred to as generalised twists and relate the problem to extremising an associated energy on the compact Lie group SO(n). The main result is a surprising discrepancy between even and odd dimensions. In even dimensions the latter system of equations admits infinitely many smooth solutions amongst such maps whereas in odd dimensions this number reduces to one.