Restricted sumset
In additive number theory and combinatorics, a restricted sumset has the form where formula_2 are finite nonempty subsets of a field "F" and formula_3 is a polynomial over "F". If formula_4 is a constant non-zero function, for example formula_5 for any formula_6, then "S" is the usual sumset formula_7 which is denoted by "nA" if formula_8. When "S" is written as formula_10 which is denoted by formula_11 if formula_8. Note that |"S"| > 0 if and only if there exist formula_13 with formula_14. 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 formula_15 we have the inequality where formula_17, i.e. we're using modular arithmetic. We may use this to deduce the Erdős–Ginzburg–Ziv theorem: given any sequence of 2"n"−1 elements in the cyclic group formula_18, 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 formula_15, every element of formula_15  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 formula_21 if "p" is a prime and "A" is a nonempty subset of the field Z/"p"Z. 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 formula_23 be a polynomial over a field "F". Suppose that the coefficient of the monomial formula_24 in formula_23 is nonzero and formula_26 is the total degree of formula_23. If formula_28 are  finite subsets of "F" with formula_29 for formula_30, then there are formula_13 such that formula_32. 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.