How to find groups? (and how to use them in Erdős geometry?)
 György Elekes,
 Endre Szabó
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Geometric questions which involve Euclidean distances often lead to polynomial relations of type F(x, y, z)=0 for some F ∈ ℝ[x, y, z]. Several problems of Combinatorial Geometry can be reduced to studying such polynomials which have many zeroes on n×n×n Cartesian products. The special case when the relation F = 0 can be rewritten as z = f(x, y), for a polynomial or rational function f ∈ ℝ(x, y), was considered in [8]. Our main goal is to extend the results found there to full generality (and also to show some geometric applications, e.g. one on “circle grids”).
The main result of our paper concerns lowdegree algebraic sets F which contain “too many” points of a (large) n×n×n Cartesian product. Then we can conclude that, in a neighborhood of almost any point, the set F must have a very special (and very simple) form. More precisely, then either F is a cylinder over some curve, or we find a group behind the scene: F must be the image of the graph of the multiplication function of an appropriate algebraic group (see Theorem 3 for the 3D special case and Theorem 27 in full generality).
 Title
 How to find groups? (and how to use them in Erdős geometry?)
 Journal

Combinatorica
Volume 32, Issue 5 , pp 537571
 Cover Date
 201205
 DOI
 10.1007/s0049301225056
 Print ISSN
 02099683
 Online ISSN
 14396912
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 05A16
 14N10
 Industry Sectors
 Authors

 György Elekes ^{(1)}
 Endre Szabó ^{(2)}
 Author Affiliations

 1. Mathematical Institute of Eötvös University, Budapest, Hungary
 2. Alfréd Rényi Mathematical Institute, Budapest, Hungary