where are finite nonempty subsets of a field F and is a polynomial over F.
When , S is the usual sumset which is denoted by nA if ; when
S is written as which is denoted by if . Note that |S| > 0 if and only if there exist with .
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 Z/pZ we have the inequality
A direct consequence of the Cauchy-Davenport theorem is: Given any set S of p−1 or more elements, not necessarily distinct, of Z/pZ, every element of Z/pZ can be written as the sum of the elements of some subset (possibly empty) of S.
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.
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 F. Suppose that the coefficient of the monomial in is nonzero and is the total degree of . If are finite subsets of F 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.
- Nathanson (1996) p.44
- Geroldinger & Ruzsa (2009) pp.141–142
- Nathanson (1996) p.48
- Geroldinger & Ruzsa (2009) p.53
- Wolfram's MathWorld, Cauchy-Davenport Theorem, http://mathworld.wolfram.com/Cauchy-DavenportTheorem.html, accessed 20 June 2012.
- Geroldinger & Ruzsa (2009) p.143
- Nathanson (1996) p.77
- Dias da Silva, J. A.; Hamidoune, Y. O. (1994). "Cyclic spaces for Grassman derivatives and additive theory". Bulletin of the London Mathematical Society 26 (2): 140–146. doi:10.1112/blms/26.2.140.
- Alon, Noga; Nathanson, Melvyn B.; Ruzsa, Imre (1996). "The polynomial method and restricted sums of congruence classes" (PDF). Journal of Number Theory 56 (2): 404–417. doi:10.1006/jnth.1996.0029. MR 1373563.
- Hou, Qing-Hu; Sun, Zhi-Wei (2002). "Restricted sums in a field". Acta Arithmetica 102 (3): 239–249. doi:10.4064/aa102-3-3. MR 1884717.
- Károlyi, Gyula (2004). "The Erdős–Heilbronn problem in abelian groups". Israel Journal of Mathematics 139: 349–359. doi:10.1007/BF02787556. MR 2041798.
- Alon, Noga (1999). "Combinatorial Nullstellensatz" (PDF). Combinatorics, Probability and Computing 8 (1–2): 7–29. doi:10.1017/S0963548398003411. MR 1684621.
- Alon, Noga; Tarsi, Michael (1989). "A nowhere-zero point in linear mappings". Combinatorica 9 (4): 393–395. doi:10.1007/BF02125351. MR 1054015.
- Geroldinger, Alfred; Ruzsa, Imre Z., eds. (2009). Combinatorial number theory and additive group theory. Advanced Courses in Mathematics CRM Barcelona. Elsholtz, C.; Freiman, G.; Hamidoune, Y. O.; Hegyvári, N.; Károlyi, G.; Nathanson, M.; Solymosi, J.; Stanchescu, Y. With a foreword by Javier Cilleruelo, Marc Noy and Oriol Serra (Coordinators of the DocCourse). Basel: Birkhäuser. ISBN 978-3-7643-8961-1. Zbl 1177.11005.
- Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics 165. Springer-Verlag. ISBN 0-387-94655-1. Zbl 0859.11003.