Linear extension of the Erdos-Heilbronn conjecture

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:119

Article

Sun, Zhi-Wei¹  Zhao, Li-Lu² 

[1] Nanjing Univ, Dept Math, Nanjing 210093, Peoples R China

[2] Univ Hong Kong, Dept Math, Hong Kong, Hong Kong, Peoples R China

关键词: Combinatorial Nullstellensatz;    Erdos-Heilbronn conjecture;    Linear extension;    Value sets of polynomials over a field;   

DOI  :  10.1016/j.jcta.2011.09.003

来源: Elsevier

【 摘 要 】

The famous Erclos-Heilbronn conjecture plays an important role in the development of additive combinatorial number theory. In 2007 Z.W. Sun made the following further conjecture (which is the linear extension of the Erclos-Heilbronn conjecture): For any finite subset A of a field F and nonzero elements a(1), ..., a(n) of F, we have [{a(1)x(1) + ... + a(n)x(n): x(1,) (...,)x(n) epsilon A, and xi not equal x(j) if i not equal j}] >= minip(F)- 8, n(lAl - n) + 1}, where the additive order p(F) of the multiplicative identity of F is different from n + 1, and delta epsilon {0, 1} takes the value 1 if and only if n = 2 and a(1) + a(2) = 0. In this paper we prove this conjecture of Sun when p(F) >= n(3n- 5)/2. We also obtain a sharp lower bound for the cardinality of the restricted sumset {x(1) + ... + X(n): X(1) epsilon A(1), ... X(n) epsilon A(n), and P(x(1), ...,x(n)) not equal 0 0}, where A(1) ,..., A(n) are finite subsets of a field F and P(x(1) ,..., x(n)) is a general polynomial over F. (C) 2011 Elsevier Inc. All rights reserved.

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