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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

Pham, Thang; Vinh, Le; de Zeeuw, Frank
Combinatorica , Volume 39 (2) – Jun 5, 2018
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16 pages

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Publisher
Springer Journals
Copyright
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

Combinatorica – Springer Journals

Published: Jun 5, 2018

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