# Three-Variable Expanding Polynomials and Higher-Dimensional Distinct Distances

Three-Variable Expanding Polynomials and Higher-Dimensional Distinct Distances Combinatorica 16pp. COMBINATORICA DOI: 10.1007/s00493-017-3773-y Bolyai Society { Springer-Verlag THREE-VARIABLE EXPANDING POLYNOMIALS AND HIGHER-DIMENSIONAL DISTINCT DISTANCES THANG PHAM, LE ANH VINH, FRANK DE ZEEUW Received February 24, 2017 Revised August 25, 2017 We determine which quadratic polynomials in three variables are expanders over an arbi- trary eld F. More precisely, we prove that for a quadratic polynomial f2F[x;y;z], which 3=2 is not of the form g(h(x)+k(y)+l(z)), we have jf (ABC )jN for any sets A;B;CF with jAj =jBj =jCj =N , with N not too large compared to the characteristic of F. We give several related proofs involving similar ideas. We obtain new lower bounds on 2 2 2 d jA+A j and maxfjA+Aj;jA +A jg, and we prove that a Cartesian product AAF determines almost jAj distinct distances if jAj is not too large. 1. Introduction Let F be an arbitrary eld. We use the convention that if F has positive characteristic, we denote the characteristic by p, while if F has characteristic 2=3 zero, we set p =1. Thus, a condition like N <p is restrictive in positive characteristic, but vacuous in characteristic zero. Although our focus is on nite elds, some of our results may be http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Combinatorica Springer Journals

# Three-Variable Expanding Polynomials and Higher-Dimensional Distinct Distances

, Volume OnlineFirst – Jun 5, 2018
16 pages

/lp/springer_journal/three-variable-expanding-polynomials-and-higher-dimensional-distinct-DuhfY84WEZ
Publisher
Springer Berlin Heidelberg
Copyright © 2018 by János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Combinatorics; Mathematics, general
ISSN
0209-9683
eISSN
1439-6912
D.O.I.
10.1007/s00493-017-3773-y
Publisher site
See Article on Publisher Site

### Abstract

Combinatorica 16pp. COMBINATORICA DOI: 10.1007/s00493-017-3773-y Bolyai Society { Springer-Verlag THREE-VARIABLE EXPANDING POLYNOMIALS AND HIGHER-DIMENSIONAL DISTINCT DISTANCES THANG PHAM, LE ANH VINH, FRANK DE ZEEUW Received February 24, 2017 Revised August 25, 2017 We determine which quadratic polynomials in three variables are expanders over an arbi- trary eld F. More precisely, we prove that for a quadratic polynomial f2F[x;y;z], which 3=2 is not of the form g(h(x)+k(y)+l(z)), we have jf (ABC )jN for any sets A;B;CF with jAj =jBj =jCj =N , with N not too large compared to the characteristic of F. We give several related proofs involving similar ideas. We obtain new lower bounds on 2 2 2 d jA+A j and maxfjA+Aj;jA +A jg, and we prove that a Cartesian product AAF determines almost jAj distinct distances if jAj is not too large. 1. Introduction Let F be an arbitrary eld. We use the convention that if F has positive characteristic, we denote the characteristic by p, while if F has characteristic 2=3 zero, we set p =1. Thus, a condition like N <p is restrictive in positive characteristic, but vacuous in characteristic zero. Although our focus is on nite elds, some of our results may be

### Journal

CombinatoricaSpringer Journals

Published: Jun 5, 2018

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