where are finite nonempty subsets of a field and is a polynomial over .
When , is the usual sumset which is denoted by if ; when
is written as which is denoted by if . Note that if and only if there exist with .
The Erdős–Heilbronn conjecture posed by Paul Erdős and Hans Heilbronn in 1964 states that if is a prime and is a nonempty subset of the field . This was first confirmed by J.A. Dias da Silva and Y.O. Hamidoune in 1994 who showed that
where is a finite nonempty subset of a field , and is a prime if is of characteristic , and if is of characteristic 0. Various extensions of this result were given by Noga Alon, M. B. Nathanson and I. Ruzsa in 1996, Q. H. Hou and Zhi-Wei Sun in 2002, and G. Karolyi in 2004.
A powerful tool in the study of lower bounds for cardinalities of various restricted sumsets is the following fundamental principle: the combinatorial Nullstellensatz Let be a polynomial over a field . Suppose that the coefficient of the monomial in is nonzero and is the total degree of . If are finite subsets of with for , then there are such that .
The method using the combinatorial Nullstellensatz is also called the polynomial method. This tool was rooted in a paper of N. Alon and M. Tarsi in 1989, and developed by Alon, Nathanson and Ruzsa in 1995-1996, and reformulated by Alon in 1999.
| doi = 10.1017/S0963548398003411}}
| doi = 10.1006/jnth.1996.0029}}
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