{"document_url": "https://en.wikipedia.org/wiki/Restricted_sumset"}
Sumset of a field subject to a specific polynomial restriction

In additive number theory and combinatorics, a restricted sumset has the form
where
are finite nonempty subsets of a field F and
is a polynomial over F.

If
is a constant non-zero function, for example
for any
, then
is the usual sumset
which is denoted by
if
When
S is written as
which is denoted by
if
Note that |S| > 0 if and only if there exist
with
Cauchy–Davenport theorem

The Cauchy–Davenport theorem, named after Augustin Louis Cauchy and Harold Davenport, asserts that for any prime p and nonempty subsets A and B of the prime order cyclic group
we have the inequality
where
, i.e. we're using modular arithmetic.

We may use this to deduce the Erdős–Ginzburg–Ziv theorem: given any sequence of 2n−1 elements in the cyclic group
, there are n elements that sum to zero modulo n. (Here n does not need to be prime.)

A direct consequence of the Cauchy-Davenport theorem is: Given any sequence S of p−1 or more nonzero elements, not necessarily distinct, of
, every element of
can be written as the sum of the elements of some subsequence (possibly empty) of S.

Kneser's theorem generalises this to general abelian groups.

Erdős–Heilbronn conjecture

The Erdős–Heilbronn conjecture posed by Paul Erdős and Hans Heilbronn in 1964 states that
if p is a prime and A is a nonempty subset of the field Z/pZ. This was first confirmed by J. A. Dias da Silva and Y. O. Hamidoune in 1994 who showed that
where A is a finite nonempty subset of a field F, and p(F) is a prime p if F is of characteristic p, and p(F) = ∞ if F 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.

Combinatorial Nullstellensatz

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.