Capturing forms in dense subsets of finite fields

Brandon Hanson. "Capturing forms in dense subsets of finite fields." Acta Arithmetica 160.3 (2013): 277-284. <http://eudml.org/doc/278916>.

@article{BrandonHanson2013,
abstract = {An open problem of arithmetic Ramsey theory asks if given an r-colouring c:ℕ → 1,...,r of the natural numbers, there exist x,y ∈ ℕ such that c(xy) = c(x+y) apart from the trivial solution x = y = 2. More generally, one could replace x+y with a binary linear form and xy with a binary quadratic form. In this paper we examine the analogous problem in a finite field $_q$. Specifically, given a linear form L and a quadratic form Q in two variables, we provide estimates on the necessary size of $A ⊂ _q$ to guarantee that L(x,y) and Q(x,y) are elements of A for some $x,y ∈ _q$.},
author = {Brandon Hanson},
journal = {Acta Arithmetica},
keywords = {finite fields; linear; quadratic; forms; dense; Ramsey},
language = {eng},
number = {3},
pages = {277-284},
title = {Capturing forms in dense subsets of finite fields},
url = {http://eudml.org/doc/278916},
volume = {160},
year = {2013},
}

TY - JOUR
AU - Brandon Hanson
TI - Capturing forms in dense subsets of finite fields
JO - Acta Arithmetica
PY - 2013
VL - 160
IS - 3
SP - 277
EP - 284
AB - An open problem of arithmetic Ramsey theory asks if given an r-colouring c:ℕ → 1,...,r of the natural numbers, there exist x,y ∈ ℕ such that c(xy) = c(x+y) apart from the trivial solution x = y = 2. More generally, one could replace x+y with a binary linear form and xy with a binary quadratic form. In this paper we examine the analogous problem in a finite field $_q$. Specifically, given a linear form L and a quadratic form Q in two variables, we provide estimates on the necessary size of $A ⊂ _q$ to guarantee that L(x,y) and Q(x,y) are elements of A for some $x,y ∈ _q$.
LA - eng
KW - finite fields; linear; quadratic; forms; dense; Ramsey
UR - http://eudml.org/doc/278916
ER -