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

Three-Variable Expanding Polynomials and Higher-Dimensional Distinct Distances We determine which quadratic polynomials in three variables are expanders over an arbitrary field \$\$\mathbb{F}\$\$ F . More precisely, we prove that for a quadratic polynomial f ∈ \$\$\mathbb{F}\$\$ F [x,y,z], which is not of the form g(h(x)+k(y)+l(z)), we have |f(A×B×C)|≫N 3/2 for any sets A,B,C ⊂ \$\$\mathbb{F}\$\$ F with |A|=|B|=|C|=N, with N not too large compared to the characteristic of F. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Combinatorica Springer Journals

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

, Volume 39 (2) – Jun 5, 2018
16 pages      /lp/springer_journal/three-variable-expanding-polynomials-and-higher-dimensional-distinct-DuhfY84WEZ
Publisher
Springer Journals
Copyright © 2018 by János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Combinatorics; Mathematics, general
ISSN
0209-9683
eISSN
1439-6912
DOI
10.1007/s00493-017-3773-y
Publisher site
See Article on Publisher Site

### Abstract

We determine which quadratic polynomials in three variables are expanders over an arbitrary field \$\$\mathbb{F}\$\$ F . More precisely, we prove that for a quadratic polynomial f ∈ \$\$\mathbb{F}\$\$ F [x,y,z], which is not of the form g(h(x)+k(y)+l(z)), we have |f(A×B×C)|≫N 3/2 for any sets A,B,C ⊂ \$\$\mathbb{F}\$\$ F with |A|=|B|=|C|=N, with N not too large compared to the characteristic of F.

### Journal

CombinatoricaSpringer Journals

Published: Jun 5, 2018

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